# The Prime Mind — Core Research Journal and Technical Handoff

## 1. Read this first

The project makes the multiplicative structure of integers visible. Each integer has a unique prime factorization; primes are the elementary building blocks, and composites are combinations of those blocks. We arrange those combinations into factor layers, largest-prime-factor branches, divisor clocks, and discrete wave-like signals. Linked views let someone move between the actual integer, its factorization, its geometry, and the equation describing it.


There are four central mathematical connections:

1. **Branch geometry:** a largest-prime-factor branch has exact lower and upper bounds at each factor layer. The retained geometric-law lab calls this the *Terminal-branch envelope law*. A statement and proof appear below.
2. **Composite coverage:** divisor clocks and canonical largest-prime-factor products can independently cover precisely the composites in a suitable future window using only information from behind a frozen boundary. Their uncovered positions are primes.
3. **Prime powers and zeta zeros:** the classical explicit formula connects weighted prime-power events with a sum over zeta zeros. The current app visualizes a finite version of that formula and lets users inspect individual zero-pair contributions.
4. **Prime Phase Field prediction program:** for a bounded interval, primality is exactly the zero set of the residue-phase field generated by the required earlier prime clocks. This exact construction is a segmented sieve written in clock/wave language. The active research question is whether that high-dimensional field has a smaller geometric or spectral summary that can rank the next prime better than matched arithmetic baselines.

**Current checkpoint (2026-09-08):** the prediction program is recorded in §§12–21. The latest completed result is the first prospectively frozen finite-zeta threshold validation at \(S=70{,}001\), \(H=500\), and one million positive zero ordinates, recorded in §21. The earlier audited high-scale partial-bank run at \(S=459{,}824{,}117\), \(H=5{,}000\), and \(q\le97\) is recorded in §19. The companion draft is `Prime Phase Field - Working Paper v0.1.md`. “Prime Phase Field” is a project working name, not a claim of historical novelty.

### Current live handoff

The exact full clock bank remains a bounded sieve: its zero set is exactly the primes in the tested interval. The new \(q\le97\) experiment shows that only 25 earlier prime clocks reduce 5,000 future integers to 600 survivors while eliminating 92.7487% of composites. All 256 primes survive, alongside 344 false survivors. The true next prime is survivor rank 2 in the opening gap, and across 256 completed gaps every next prime lies within the first 10 survivors.

The critical resolved point is that **candidate-local higher-power values cannot rank survivors**. A survivor of all base clocks \(q\le97\) automatically has zero hits in every associated \(q^a\) lane. Therefore the next research target is not the valuation at the candidate itself, but a frozen ranking rule built from the surrounding phase field, distances to strikes, neighborhood geometry, past-only branch/gap summaries, or a separately evaluated local zeta score. Any such rule must beat the ordinary wheel/geometric survivor baseline on a genuinely new interval.

The exact identities are solid. The plots are representations of those identities, and the measured patterns are experimental observations. The first prospectively frozen zeta-score rule has now succeeded perfectly on one untouched 500-integer interval: the predeclared threshold classified all 50 future primes and all 450 composites correctly. This is a genuine out-of-sample numerical validation at one scale, not yet a general theorem or an efficiency result. **We have not established a universal next-prime shortcut, a proof of the Riemann hypothesis, or the historical novelty of the branch-envelope theorem.** What is now established is an exact bounded prime-hole field, a rigorous frozen protocol for testing compression, and one successful frozen finite-zeta validation that must be replicated across disjoint windows and increasing scales.

## 2. Notation to keep consistent

LPF means **largest prime factor throughout this project**. Some mathematical software uses similar abbreviations for the least prime factor; that is not our convention.

| Symbol | Meaning |
|---|---|
| \(n\) | An integer being displayed or tested; normally \(n\ge2\). |
| \(p=\operatorname{LPF}(n)\) | The largest prime factor, which owns the canonical branch. |
| \(\operatorname{SPF}(n)\) | The smallest prime factor. |
| \(\Omega(n)\) | Number of prime factors **with multiplicity**. For \(12=2^2\cdot3\), this is 3. |
| \(\omega(n)\) | Number of **distinct** prime factors. For 12, this is 2. |
| \(L\) | Factor layer, usually \(L=\Omega(n)\). Primes occupy layer 1. |
| \(u=n/p\) | Canonical LPF cofactor: remove one occurrence of the largest prime. |
| \(q\) | A prime acting as a divisor clock; it need not be the LPF. |
| \(a\), \(v_q(n)\) | Power exponent, and the exponent of \(q\) in \(n\). |
| \(S\) | The fixed starting prime / freeze boundary for a run. |
| \(H\), \(N=S+H\) | Run horizon and upper endpoint. |
| \(P_s,P\) | Consecutive primes bounding one gap. \(P_s\) changes as we move through the run; \(S\) does not. |
| \(g=P-P_s\) | Prime-gap length. The interior contains \(g-1\) integers. |
| \(b_0=P-1\), \(h=(P-1)/2\) | Final pre-prime integer, and its half-coordinate, for odd \(P\). |
| \(b_q(P)\) | Last multiple of clock prime \(q\) before \(P\). |
| \(r_q=b_q/q\) | Clock quotient. Older exports sometimes called this \(m\). |
| \(\delta_q=P-b_q\), \(\phi_q=\delta_q/q\) | Distance from that clock landing to the prime, and normalized phase. |
| \(p_{\max}\) | Largest LPF owner represented anywhere inside a particular prime gap. |
| \(b^*\) | Landing owned by \(p_{\max}\); choose the latest landing if there is a tie. |
| \(k=b^*/p_{\max}\), \(d=P-b^*\) | Selected-owner multiplier and its remaining distance to \(P\). |
| \(\Lambda(n)\), \(\psi(x)\) | Von Mangoldt prime-power weight and its cumulative sum; defined in §8. |
| \(\rho=\tfrac12+i\gamma\), \(K\) | A tabulated critical-line zero and number of positive zero ordinates retained. |
| \(Q_N\) | The primes \(q\le\sqrt N\) required for exact primality coverage through \(N\). |
| \(\Phi_{Q_N}(S)\) | The seed's clock-phase fingerprint \(((S\bmod q))_{q\in Q_N}\). |
| \(\tau_q(S)\) | Distance from \(S\) to the next \(q\)-clock strike: \((-S)\bmod q\). |
| \(\mathcal H_N(n)\) | Proper-divisor clock-hit count. The app often labels the same idea \(H(n)\); calligraphic \(\mathcal H\) avoids confusion with horizon \(H\). |
| \(\mathcal W_N(n)\) | Nested prime-power hit height, including the base \(q\)-lanes. |
| \(B\) | A deliberately incomplete small-prime cutoff in a partial-sieve experiment. |
| \(P_B(n)\) | Indicator that \(n\) survives all proper-divisor clocks \(q\le B\). |
| \(\widehat\Lambda_K(n)\) | Local prime-power estimate obtained by first-differencing a finite zeta-zero reconstruction. |
| \(R_{B,K}(n)\) | Proposed hybrid score: partial-sieve survival multiplied by the normalized local zeta estimate. |

Do not use one unqualified \(m\) for the clock quotient, LPF cofactor, selected-owner multiplier, and half-coordinate. They can coincide in an example but describe different objects. Also distinguish an \(\Omega\)-layer, a factor’s lateral rank in a skeleton, and a \(q^a\)-clock lane: their colors do not mean the same thing.

The integer 1 is a separate empty-factorization case: \(\Omega(1)=0\), and it has no largest prime factor. A code sentinel such as LPF(1)=1 is only a convention.

## 3. The mathematical skeleton and branch theorem

### Canonical ownership

Write the prime factorization as

\[
n=\prod_p p^{v_p(n)},\qquad \Omega(n)=\sum_p v_p(n).
\]

The branch belonging to prime \(p\) consists of integers whose largest prime factor is \(p\):

\[
\mathcal B_p=\{p\,u:\text{every prime factor of }u\text{ is at most }p\}.
\]

Here \(u=1\) includes the prime root itself. Unique factorization gives every integer \(n\ge2\) exactly one owner. Other clocks may strike the same integer, but those overlaps do not create extra canonical owners.

When a tree edge is meant to retain this terminal branch, remove the **smallest** prime factor:

\[
90\to45\to15\to5.
\]

Every node remains in the LPF-5 branch. This operation differs from the support cofactor \(n/\operatorname{LPF}(n)\), which is 18 for \(n=90\).

### Terminal-branch envelope theorem

This is the formal law explicitly recoverable from `prime-branch-envelope-geometric-law.html`. Treat the name as a project label; its historical novelty has not been established.

For a prime \(p\) and an integer layer \(L\ge1\), define

\[
\mathcal B_{p,L}=\{n:\operatorname{LPF}(n)=p,\ \Omega(n)=L\}.
\]

Then every member satisfies

\[
\boxed{p\,2^{L-1}\le n\le p^L,}
\]

and both bounds are attained.

**Proof.** Remove one occurrence of the largest prime \(p\). The remaining \(L-1\) prime factors are each at least 2 and at most \(p\). Their product therefore lies between \(2^{L-1}\) and \(p^{L-1}\). Multiplication by \(p\) gives the bounds. The integers \(p2^{L-1}\) and \(p^L\) have the required LPF and factor count, so the bounds are sharp. This also covers \(L=1\), when both equal \(p\).

On logarithmic axes the envelope becomes

\[
\log p+(L-1)\log2\le\log n\le L\log p.
\]

This gives exact straight boundaries when plotting log-size against factor layer. It does **not** fill every integer between them. For \(p=5,L=2\), the members are only \(10,15,25\).

Useful consequences, all derivable directly from the same factorization argument:

- Layer \(L\) first appears in branch \(p\) at \(p2^{L-1}\).
- Through an integer endpoint \(N\ge p\), that branch’s greatest depth is \(1+\lfloor\log_2(N/p)\rfloor\).
- Across all integers \(2\le n\le N\), the greatest depth is \(\lfloor\log_2N\rfloor\), attained by a power of 2.
- For a single prime \(q\), the greatest possible exponent through \(N\) is \(\lfloor\log_qN\rfloor\).
- A whole, untruncated branch layer has \(\binom{\pi(p)+L-2}{L-1}\) members: choose a multiset of \(L-1\) factors from the \(\pi(p)\) primes up to \(p\), then append one \(p\). An endpoint cutoff generally reduces this count.

These statements explain different kinds of “growth.” A new factor-depth record, a new \(q\)-power lane, and more occupied points in an existing layer are distinct events.

## 4. Divisor clocks, prime-power waves, and exact future coverage

### What a clock measures

The raw divisibility signal for prime \(q\) is

\[
D_q(n)=\mathbf1_{q\mid n}.
\]

It repeats every \(q\) integers. A power lane \(D_{q^a}(n)\) repeats every \(q^a\). Nested power lanes recover multiplicity:

\[
v_q(n)=\sum_{a\ge1}\mathbf1_{q^a\mid n},\qquad
\Omega(n)=\sum_{q\ \mathrm{prime}}v_q(n).
\]

The lifted-copy identity is exact even if \(q\) also divides \(m\):

\[
\boxed{\Omega(q^a m)=a+\Omega(m).}
\]

This is one precise reason repeated structure appears on different power layers. “Frequency” here means repetitions per integer, such as \(1/q^a\). The primes themselves are not a single equally spaced fundamental frequency. A single sine wave does not describe the entire field, but sums of periodic functions are perfectly legitimate descriptions.

A fixed finite collection of raw power lanes repeats after their least common multiple. If the retained exponents are \(a_q\), a common period is \(\prod_q q^{a_q}\). Introducing new primes or deeper powers enlarges that period. The growing, unlimited system should not be confused with one fixed repeating bank.

For primality coverage through \(N\), all primes \(q\le\sqrt N\) suffice. Exclude a prime’s self-hit:

\[
C_q(n)=\mathbf1_{q\mid n\ \mathrm{and}\ q<n}.
\]

For \(2\le n\le N\), \(\sum_q C_q(n)=0\) if and only if \(n\) is prime. Every composite has a prime divisor at most its square root. A conventional sieve can instead activate clock \(q\) at \(q^2\); it gives the same composite union, although its individual strike counts differ.

The app’s all-clock power sum uses a finite bank \(Q\):

\[
V_Q(n)=\sum_{q\in Q}v_q(n).
\]

This need not equal \(\Omega(n)\): a large prime factor can lie outside \(Q\). Zero-testing also needs the self-hit convention above. These distinctions matter particularly in runs beginning near 2; see the concrete implementation note in §11.

### Frozen-frontier coverage theorem

The frontier experiments use a fixed prime boundary \(S\) and a future endpoint \(N=S+H<2S\). Assume the primes and factorizations through \(S\) are known.

Every future composite \(S<n\le N\) has one canonical representation

\[
n=p\,u,\qquad p=\operatorname{LPF}(n),\quad u\ge2,
\]

with every prime factor of \(u\) at most \(p\). Moreover,

\[
p\le n/2<S,\qquad u\le n/2<S.
\]

Thus both the branch prime and cofactor are already in the known region. Enumerating these admissible products covers every future composite once and covers no primes. Independently, the required small-prime clocks cover exactly the same composite positions, possibly with overlaps.

That proves the masks agree and that the first uncovered integer is the next prime **if it occurs within the examined window**. An empty list of holes does not license calling the integer after the window prime.

This is an exact sieve construction derived from factorization. It does not establish a faster-than-sieving prediction rule. The frontier lab’s **Freeze → Reveal** design separates the derived mask from later truth-checking. The wavefield lab, which factors the displayed integers to study their structure, is a descriptive tool and does not inherit that experimental separation automatically.

## 5. How the pictures encode the numbers

### Earlier logarithmic branch wall

The earlier wall geometry uses

\[
x(n)=\frac{\log\operatorname{LPF}(n)}{\log N},\qquad
z(n)=\frac{\log n}{\log N},\qquad L(n)=\Omega(n).
\]

LPF selects the branch, log-size selects height, and factor count identifies the layer. Thus \(10,15,25\) belong to the LPF-5 column and layer 2, at three different log-heights. Do not spread them sideways and present the invented offsets as original coordinates.

### Current factor skeleton and its collapsed side view

The newer skeleton is another representation. It has an integer-time direction, a lateral factor-rank direction, and log-factor height. Sort prime-factor occurrences from largest downward. Put one LPF occurrence in the center, then alternate the remaining occurrences in slots \(+1,-1,+2,-2,\ldots\). Repeated factors remain separate occurrences.

The 2D stacked side view drops the time direction while preserving those lateral slots and heights. It overlays **whole skeletons**; it does not lay the individual branches out end-to-end. Grouping 2, 5, 10, or 20 gaps overlays more of these complete units in a common projection. Shared prime endpoints should be counted once.

Some projected points necessarily coincide. Multiplicity indicators, transparency, and inspection details should expose that count. A claimed “no hidden factors” view must preserve the data even when several factors share a pixel. Variable dot sizes are display encodings, not an extra mathematical coordinate. The alternating placement itself introduces visual symmetry; that symmetry is a construction choice.

### Which height is a wave?

Adding every factor’s logarithm gives \(\log n\). That is an identity and produces a size trend. To study composite support separately, remove one largest-factor occurrence:

\[
A_{\log}(n)=\sum_{\text{remaining occurrences}}\log p
=\log\frac{n}{\operatorname{LPF}(n)},\qquad
A_{\Omega}(n)=\Omega(n)-1.
\]

Both are nonnegative and vanish exactly at primes for \(n\ge2\). Consequently a prime gap appears as composite support between two zero endpoints. Calling this a “wave” is useful descriptive language. It is not evidence of a physical field, and knowing the factorization to calculate the zero is not independent prime prediction.

By contrast, \(\omega(n)-1\) also vanishes at composite prime powers such as 8 and 25. It measures distinct-branch complexity, not all compositeness. An aggregate called “energy” in the app is a sum of log support, not a physical energy measurement.

## 6. Prime boundaries, landings, and ratios

For an earlier clock prime \(q<P\), the last landing before prime \(P\) is

\[
b_q(P)=P-(P\bmod q),\quad r_q=b_q/q,
\quad\delta_q=P\bmod q,\quad\phi_q=\delta_q/q.
\]

Across a prime gap \(g\), residues advance by addition modulo \(q\). A phase fingerprint is therefore an exact encoding of those clock positions. It is not by itself a compressed rule for the next uncovered integer.

For each nonempty gap interior, select

\[
p_{\max}=\max_{P_s<b<P}\operatorname{LPF}(b),\qquad
b^*=\max\{b:P_s<b<P,\operatorname{LPF}(b)=p_{\max}\}.
\]

Then

\[
\boxed{P=k\,p_{\max}+d},\qquad
\boxed{\frac{P}{p_{\max}}=k+\frac{d}{p_{\max}}}.
\]

The largest owner is numerically closest to \(P\) among the **owners represented in the gap**; its landing need not be the closest composite. Since a composite owner satisfies \(p\le b/2\), even this largest owner is below \(P/2\). Ratios near 2, 3, or other small integers often reflect the integer multiplier and small remaining distance, rather than a new independent law.

Example: \(P_s=19,P=23\).

| Integer | Factorization | \(\Omega\) | LPF | Role |
|---:|---|---:|---:|---|
| 19 | 19 | 1 | 19 | Previous prime |
| 20 | \(2^2\cdot5\) | 3 | 5 | Composite support |
| 21 | \(3\cdot7\) | 2 | 7 | Last \(q=3\) landing |
| 22 | \(2\cdot11\) | 2 | 11 | Final composite and largest-owner landing |
| 23 | 23 | 1 | 23 | New prime |

Here \(g=4,b_0=b^*=22,h=p_{\max}=11,k=2,d=1\). Clock 2 has \(r_2=11,\delta_2=1,\phi_2=1/2\); clock 3 has \(r_3=7,\delta_3=2,\phi_3=2/3\). The coincidences in this example are not general identities between all those variables.

For the gap 47 to 53, the largest owner is 17 at \(51=3\cdot17\), so \(k=3,d=2\). The final composite is separately \(52=2\cdot26\). Restricting the model to multiplier 2 loses real cases.

Every odd prime has \(P=2h+1\), but \(h\) need not be prime. Advancing an earlier branch prime and applying \(2q+1\) fails, for example, at \(q=17\), where the result is \(35=5\cdot7\). Also, the phase \(\phi_2=1/2\) holds at every odd integer; that particular half is a parity identity.

## 7. What the supplied export actually establishes

The three supplied CSVs describe \(S=61{,}729,H=500,N=62{,}229\). An independent arithmetic check of their factorizations, coverage, prime boundaries, LPF ownership, products, residues, phases, and layer identities found **zero errors in the checks performed**.

| Quantity | Checked result |
|---|---:|
| Future integer rows, 61,730 through 62,229 | 500 |
| Prime holes / composite positions | 43 / 457 |
| Complete consecutive-prime transitions | 43 |
| First new prime / last complete boundary | 61,751 / 62,219 |
| Required prime clocks through \(\sqrt{62{,}229}\) | 53 |
| Last-hit records | 2,279 = 43 × 53 |
| Twin-prime gaps | 4 |
| Prime half-coordinates \(h=(P-1)/2\) | 2 of 43 |
| Largest-owner multiplier \(k=2\) / \(k=3\) | 18 / 4 gaps |
| Greatest observed factor depth | 13 |

The waveform app may display the starting prime as well, giving 501 integers and 44 primes including the seed. This is a counting convention, not a discrepancy with the 500 future rows.

The first exported gap is particularly useful for teaching:

\[
61{,}729\to61{,}751,\qquad
b^*=61{,}742=2\cdot30{,}871,\qquad d=9.
\]

Its immediate predecessor has a much smaller owner:

\[
b_0=61{,}750=2\cdot5^3\cdot13\cdot19,\qquad \Omega(b_0)=6.
\]

Thus a boundary does not always finish with only two or three prime-factor occurrences.

We also tested the proposed “more complex in the middle” pattern. For each complete gap, put interior integers at \(t=(n-P_s)/g\). Compare the middle third \(1/3<t<2/3\) with the outer thirds, excluding prime endpoints. Only 39 gaps have both groups populated.

| Raw metric | Gaps with higher middle mean | Gaps with lower middle mean | Pooled middle-minus-outer mean |
|---|---:|---:|---:|
| \(\Omega(n)-1\) | 16 | 23 | −0.1782 |
| \(\log(n/\operatorname{LPF}(n))\) | 17 | 22 | −0.3279 |

These results do not support a universal central rise or terminal taper. They do not erase the visible repeated structures; they show why the metric and control matter. Fixed zero endpoints, parity, smoothing, and stretching each gap to unit width can themselves create convincing wave shapes. The app includes a limited control that shuffles interior heights among positions of the same parity within each gap. Treat it as a descriptive comparison, not a significance proof.

## 8. The precise connection to zeta zeros

### Which arithmetic signal connects?

The direct classical bridge uses **prime-power events**, not the sum of every composite’s factors:

\[
\Lambda(n)=
\begin{cases}
\log p,&n=p^a,\ p\text{ prime},\ a\ge1,\\
0,&\text{otherwise},
\end{cases}
\qquad
\psi(x)=\sum_{n\le x}\Lambda(n).
\]

For example, 6 is struck by clocks 2 and 3 but has \(\Lambda(6)=0\); 8 has \(\Lambda(8)=\log2\). The explicit formula expresses \(\psi\) using a leading term \(x\), contributions from zeta zeros, and correction terms. This is established mathematics, not a newly discovered connection in this project. See [NIST DLMF §25.16, mathematical applications](https://dlmf.nist.gov/25.16).

For \(x>1\) away from prime-power discontinuities, the conventional explicit formula is

\[
\psi(x)=x-\sum_\rho\frac{x^\rho}{\rho}
-\log(2\pi)-\frac12\log(1-x^{-2}).
\]

The sum over nontrivial zeros requires symmetric limiting order. At a discontinuity, the corresponding formula uses the midpoint of the left and right values. These qualifications matter when turning it into a numerical plot.

### What the current app computes

To avoid the jumps, use \(x=n+1/2\) and starting point \(x_0=S+1/2\). The exact arithmetic residual is

\[
E_\psi(n)=\sum_{S<m\le n}\Lambda(m)-(n-S).
\]

Using the first \(K\) positive tabulated ordinates, with \(\rho_j=1/2+i\gamma_j\), the app constructs

\[
E_K(n)=
-2\sum_{j=1}^K
\operatorname{Re}\!\left(\frac{x^{\rho_j}-x_0^{\rho_j}}{\rho_j}\right)
-\frac12\log\!\left(\frac{1-x^{-2}}{1-x_0^{-2}}\right).
\]

The constant term cancels on subtracting the start value. This finite expression approximates the residual; it is not the complete infinite zero sum. The current file embeds 1,000 positive zero ordinates, with controls retaining 10, 29, 100, 500, or 1,000 pairs. Its numerical behavior depends on the interval, truncation, and smoothing.

The key identity is

\[
x^{1/2+i\gamma}=\sqrt{x}\,e^{i\gamma\log x}.
\]

Thus a zero-pair contribution oscillates in **logarithmic position**, with angular frequency \(\gamma\) in \(\log x\) and amplitude growing like \(\sqrt x\). This differs from the integer-coordinate period \(q^a\) of a divisibility clock. A zero is not simply a clock prime under another name.

When selecting one zero, the green curve is

\[
C_j(n)=-2\operatorname{Re}\!\left(\frac{x^{\rho_j}-x_0^{\rho_j}}{\rho_j}\right).
\]

It contributes throughout the window. Before smoothing, its bounds are

\[
c_j\pm\frac{2\sqrt{x}}{\sqrt{1/4+\gamma_j^2}},\qquad
c_j=2\operatorname{Re}\left(\frac{x_0^{\rho_j}}{\rho_j}\right).
\]

Those bounds are the meaningful “half-exponent” guides. When smoothing is selected, apply the same averaging to the contribution and guides. A selected zero outside the current \(K\) is inspectable but must be labeled as excluded from the yellow sum.

### What the zero markers mean

The critical line is \(\operatorname{Re}s=1/2\); the Riemann hypothesis asserts that **all** nontrivial zeta zeros lie on it. Showing a finite known set does not prove that assertion. See [NIST DLMF §25.10, zeros](https://dlmf.nist.gov/25.10).

The current upper charts plot tabulated zeros at \((1/2,\gamma_j)\), and those same ordinates on a symbolic zero-height baseline. They do **not** independently evaluate \(\zeta(1/2+it)\) to find crossings. Their alignment with that baseline is specified by the known-zero data.

The lower chart compares arithmetic residuals and zero-sum contributions on the integer axis. It is not a plot of \(\zeta(s)\), and its curves need not cross zero at \(n=\gamma_j\). The informative correlation is between the **fluctuations of the weighted prime-power count and the reconstruction using many zeros**.

Several halves must remain distinct:

| Appearance of \(1/2\) | Meaning |
|---|---|
| \(\operatorname{Re}\rho=1/2\) | Critical-line coordinate; yields the \(\sqrt x\) factor in a zero contribution. |
| \(x=n+1/2\) | Numerical choice to sample between integer jumps. |
| \(h=(P-1)/2\), \(\phi_2=1/2\) | Odd-integer and parity identities. |
| Midpoint of a normalized prime gap | Alignment choice for comparing gap shapes. |

A shared numeral does not establish a theorem linking these constructions. The visual “lopsided-wave peak explains the critical line” idea remains a conjectural intuition, not a demonstrated result.

### Interpreting agreement and the controls

The purple comparison retains zero frequencies and amplitudes but changes phases deterministically. It probes whether the correctly phased finite reconstruction follows the arithmetic curve better in that window. It is not an independent prime-prediction model or an exhaustive statistical null.

The app can display original residual units or standardized shape comparisons. Standardizing curves separately discards their amplitude differences. For a selected pair, the current shape view uses the yellow sum’s scale so its displayed size still represents a contribution to that sum. Smoothing choices include raw data and centered means of 5, 15, or 31 points.

Pearson correlation is descriptive here: samples are dependent, smoothing changes the comparison, and increasing \(K\) need not improve every finite-window correlation monotonically. Strong agreement with sufficient terms illustrates the explicit formula; it is not evidence that every plotted line must meet at every prime. Centered smoothing also uses neighboring future samples and must not be imported into a purported prediction test.

## 9. Existing app map

The current main file is **`prime-composite-wavefield-lab.html`**. It is a standalone HTML application with embedded code and data. The latest standalone reviewed for this briefing is the version containing linked zero selections, the foreground yellow reconstruction, phase-control visibility, and half-exponent guides. An older hosted copy should not automatically be treated as the same version.

| Main-app view | What it is for |
|---|---|
| Wave field | Composite support, alternate metrics, smoothing, and prime-clock lanes. |
| 3D factor skeleton | Individual factor occurrences, center LPF, adjustable camera, and previous/next gap navigation. |
| 2D stacked side | The collapsed skeleton projection, including overlays of whole groups of gaps. |
| Prime-power waves | Inspect one \(q\), its nested powers, and the exact lifted-copy identity. |
| All-clock power sum | Stack power-lane contributions across the clock bank; inspect components and dot-size encoding. |
| Prime powers ↔ zeta zeros | Known-zero maps, linked selection, exact arithmetic residual, finite reconstruction, phase control, and half-exponent guides. |
| Large-range sampler | Explore sampled factor structure through endpoints up to one billion. |
| Gap clumps + control | Align completed gaps, overlay groups, and compare with within-gap parity-preserving controls. |
| Layer births + repeats | Factor-depth records, gap trends, twin-gap context, repeated shapes, optional gap-level signal, and exact selected-gap strike counts. |
| Clock spectrum | Finite-window frequency content of chosen arithmetic signals. |

Important companion files:

| File | Role |
|---|---|
| `prime-clock-lpf-frontier-lab.html` | Clock/LPF coverage, frozen-boundary audit, consecutive-prime matrix, last-hit ledger, and boundary-variable map. |
| `prime-branch-envelope-geometric-law.html` | Named branch-envelope law and its geometric exploration. |
| `prime-layer-wall-builder-Final.v2.html` | Earlier logarithmic wall and layer geometry. |
| `prime-factor-geometry-explorer.html` | Earlier factor-geometry exploration. |
| `prime-branch-clump-wave-lab.html` | Earlier branch-clump and wave experiments. |

These companion files supply context, not a requirement to merge all interfaces into one enormous page. Other retained frontier and compression experiments exist; inspect their own definitions before quoting their results. This handoff does not certify every historical app or conversation claim.

### Larger ranges and sampling

The dense wavefield currently supports horizons up to 5,000 integers. The separate large-range view caps the number of sampled integers at approximately 50,000 and supports endpoints up to \(10^9\). For a start near 2, representative automatic skips are 20 at one million, 200 at ten million, 2,000 at one hundred million, and 20,000 at one billion.

Its rotating selection chooses one integer at a changing offset within each block. This reduces the obvious residue aliasing of always taking, for example, every tenth integer from the same start. It does not guarantee an unbiased sample for every arithmetic question.

Every retained integer is factored exactly. Omitted integers are unclassified: the sample cannot supply complete prime counts, complete gaps, or proof that no deeper layer occurred between samples. Screen-bin extrema preserve extrema among the samples, not among all skipped integers. Irregularly selected integers also cannot be fed to an ordinary equally spaced Fourier analysis without accounting for their actual positions.

## 10. Directions worth investigating next

The next-prime experiment should remain a separate page with an explicit information boundary. The present exploratory views can help generate candidate rules; they do not establish them by themselves.

Useful questions include:

- Can a compressed summary of the known clock phases locate the first uncovered position with less work than a matched sieve baseline?
- Are repeated gap shapes still unusually similar after controlling for parity, gap length, local integer scale, smoothing, and the chosen metric?
- Does a branch’s approaching envelope boundary help forecast a layer event, and is the claim about a global record, a branch-specific record, or merely occupancy in one selected window?
- Are transitions in \(p_{\max},k,d\) informative using only completed earlier gaps? Those values for the target gap cannot be used as input before that gap is known.
- How do zero-sum agreement and amplitude error vary jointly with \(K\), position, window length, and smoothing? Plot error alongside correlation.

One exact quantity helps separate a real clock effect from a visual impression. The number of \(q^a\)-multiples strictly inside a completed gap is

\[
B_{q,a}(P_s,P)=
\left\lfloor\frac{P-1}{q^a}\right\rfloor
-\left\lfloor\frac{P_s}{q^a}\right\rfloor.
\]

This is the actual integer count. The rough phase-average \((g-1)/q^a\) is a useful comparison, not an exact answer for a prime-conditioned interval. Individual lane counts also overlap and should not be added as if every strike represented a different composite.

For any prediction claim: specify the input available through \(S\), state the candidate rule before examining the next block, use later disjoint blocks for evaluation, and count the cost of deriving features. Ground-truth factorization of the target region, known next-prime boundaries, normalized target-gap positions, or centered smoothing would otherwise leak future information. Compare against straightforward arithmetic baselines before making a performance claim.

## 11. Specific correctness items to resolve before public release

These are concrete distinctions found while inspecting the current artifacts; this document does not modify the apps.

1. **Small-prime self-hits in the all-clock power view.** `buildRecord` excludes \(q=n\) from proper-divisor clock hits, but `allPowerColumn` currently adds \(v_q(n)\) without that guard. A prime inside the clock bank can therefore have a power dot. If the intended display is composite-only, apply the same guard; otherwise label it as the literal valuation sum with the startup exception. The default high-seed run does not expose this edge case.
2. **Distinct-factor zeros.** Do not describe \(\omega-1\) as vanishing only at primes. Prime powers are counterexamples.
3. **Yellow layer-record counts.** The global formula \(\max_{n\le N}\Omega(n)=\lfloor\log_2N\rfloor\) is exact. Rings produced from completed prime gaps are run-relative records. The current seed-2 caption equating their count with \(\lfloor\log_2N\rfloor-1\) needs to account for a trailing incomplete gap that may contain the newest power of 2.
4. **Known-zero provenance and labeling.** Keep the symbolic maps clearly distinguished from an actual evaluation of zeta. The embedded ordinate table should carry an independently traceable source and numerical precision before it appears in a paper or data download; that provenance was not established by this handoff’s CSV audit.
5. **Geometry, legends, and data state.** Preserve the distinction between old branch-wall coordinates and new factor-rank coordinates, between rank colors and \(\Omega\), and between a truth-informed descriptive plot and the independently frozen coverage experiment.

## 12. Prime Phase Field: the exact bounded prediction backbone

**Added 2026-09-08 after reviewing the new wavefield, factor-skeleton, clock-spectrum, and `zeta 1`–`zeta 3` screenshots.** This section records the current shared understanding and should be read before starting another prediction experiment.

### 12.1 Current end goal

The long-range objective is a structure that can predict or sharply rank future prime locations. The immediate goal is more precise:

> Starting at a known prime \(S\), continue the exact periodic state of all required earlier prime clocks, identify their common holes, and test whether the resulting high-dimensional field can be compressed into a genuinely useful next-prime predictor.

A key semantic correction is essential. “Dissecting the starting prime” does **not** mean factoring \(S\), because a prime has no nontrivial factorization. It means measuring \(S\)'s position relative to every older divisor clock. The useful state is its residue or phase fingerprint

\[
\Phi_{Q_N}(S)=\bigl(S\bmod q\bigr)_{q\in Q_N},
\qquad
Q_N=\{q:q\text{ prime},\ q\le\sqrt N\}.
\]

Equivalently, store the next-strike distances

\[
\tau_q(S)=(-S)\bmod q.
\]

The \(q\)-clock then strikes future offsets

\[
t=\tau_q(S)+j q,
\qquad j=0,1,2,\ldots.
\]

This is exactly the “continue the pattern forward” operation visible in the experimental tools.

### 12.2 Three-layer statement

#### Visual

Imagine one circular clock for each required prime \(q\). Advancing from integer \(n\) to \(n+1\) moves every hand forward by one position. The \(q\)-clock strikes whenever the hand returns to zero. A vertical stack of one or more strikes marks a composite. A location with no proper-divisor strike is a prime hole.

#### Equation

For \(2\le n\le N\), define the proper-divisor strike

\[
C_q(n)=\mathbf 1_{q\mid n\ \text{and}\ q<n}.
\]

Then define the exact clock-hit field

\[
\boxed{
\mathcal H_N(n)=\sum_{q\le\sqrt N\atop q\ \mathrm{prime}} C_q(n).
}
\]

The exact bounded prime-hole rule is

\[
\boxed{
n\text{ is prime}\quad\Longleftrightarrow\quad\mathcal H_N(n)=0.
}
\]

For a seed \(S\) and offset \(t\), this is

\[
\mathcal H_{S,N}(t)=\mathcal H_N(S+t).
\]

Provided the horizon contains a later prime, the next prime is

\[
\boxed{
p_{\mathrm{next}}(S)
=S+\min\{t\ge1:\mathcal H_{S,N}(t)=0\}.
}
\]

#### Plain-language semantics

- \(S\) is the known prime where the run begins.
- \(H\) is the chosen number of future integers to examine.
- \(N=S+H\) is the last integer in that window.
- \(q\) is one earlier prime clock.
- \(C_q(n)=1\) means that \(q\) is a proper divisor of \(n\); otherwise it is zero.
- \(\mathcal H_N(n)\) counts how many required prime clocks strike \(n\).
- Every composite through \(N\) has at least one prime divisor at most \(\sqrt N\), so a composite cannot be a zero of the complete field.
- A prime has no proper divisor clock strike, so it is a zero.

This theorem is exact, but it is a segmented sieve expressed as a phase field. It is not by itself a shortcut beyond sieving.

For the high-seed frozen-frontier setting \(N<2S\), all required primes \(q\le\sqrt N\) lie behind \(S\), so their phases are available before the target interval is revealed. In a low-start display, either retain the explicit guard \(q<n\) or start each conventional sieve clock at \(q^2\) so that a prime does not strike itself.

### 12.3 Exact wave representation of a clock

At integer \(x\), divisibility by a period \(m\) has the finite Fourier representation

\[
\boxed{
D_m(x)=\frac1m\sum_{k=0}^{m-1}
\cos\!\left(\frac{2\pi kx}{m}\right).
}
\]

For integer \(x\),

\[
D_m(x)=
\begin{cases}
1,&m\mid x,\\
0,&m\nmid x.
\end{cases}
\]

Thus, away from the small-prime self-hit startup case, the prime-hole field can be written as a finite sum of exact periodic waves:

\[
\mathcal H_N(S+t)=
\sum_{q\le\sqrt N}
\frac1q\sum_{k=0}^{q-1}
\cos\!\left(\frac{2\pi k(S+t)}{q}\right).
\]

This makes “wavefield” mathematically literal at the integer samples. The values between integers are a chosen interpolation and do not add new arithmetic information.

The clock-spectrum screenshot is consistent with this construction: its dominant finite-window lines occur near fractions such as \(1/2,1/3,1/5,1/7,1/11\), with neighboring leakage and harmonics. Those lines are expected signatures of the retained periodic clocks, not unexplained new frequencies.

### 12.4 Prime-power extension and the zero-set preservation theorem

For prime \(q\), the nested power lanes are

\[
D_q(n),\ D_{q^2}(n),\ D_{q^3}(n),\ldots,
\]

and

\[
v_q(n)=\sum_{a\ge1}D_{q^a}(n).
\]

Define a complete nested power field through \(N\) by

\[
\mathcal W_N(n)=
\sum_{q\le\sqrt N\atop q\ \mathrm{prime}}
\sum_{a\ge1\atop q^a\le N}
\mathbf1_{q^a\mid n\ \text{and}\ q<n}.
\]

Because

\[
D_{q^a}(n)\le D_q(n)
\qquad(a\ge2),
\]

a higher power can only deepen a location already struck by its base clock. Therefore, when all required base lanes are present,

\[
\boxed{
\mathcal W_N(n)=0
\quad\Longleftrightarrow\quad
\mathcal H_N(n)=0
\quad\Longleftrightarrow\quad
n\text{ is prime}.
}
\]

This is the **prime-power zero-set preservation theorem** for the project:

> Higher prime-power lanes change composite height and internal structure, but they do not create a new composite location that the corresponding base-prime lane failed to strike.

This sharply defines their possible predictive role. Base clocks determine exact prime/composite support. Higher powers may still help describe, compress, or rank the **surrounding composite terrain**, gap length, repeated local shapes, and factor depth, but they cannot directly eliminate a base-clock survivor without adding a new base prime.

For a fixed finite bank with largest retained exponent \(a_q\) for each prime, the raw lane field has period

\[
T=\operatorname{lcm}\{q^{a_q}\}=\prod_q q^{a_q}.
\]

The app's displayed repeat near \(10^{9222.9}\) is a screen-recorded example of this enormous finite-bank period; it should be regenerated from the exported bank before being quoted as an audited number. Adding new prime clocks or deeper powers enlarges \(T\), so the growing unlimited construction is not one permanently fixed periodic sequence.

## 13. What the ten new screenshots add to the record

The screenshots form one coherent hierarchy rather than ten unrelated observations:

\[
\text{integer time}
\longrightarrow
\text{prime-clock phases}
\longrightarrow
\text{composite coverage}
\longrightarrow
\text{nested power depth}
\longrightarrow
\text{factor geometry}
\longrightarrow
\Lambda,\psi
\longrightarrow
\text{zeta-zero spectrum}.
\]

The numerical values below are **screen-recorded observations**, not yet a replacement for raw CSV audits.

| Image | Recorded content and interpretation |
|---|---|
| `Screenshot 2026-09-07 062603.png` | High-seed nested \(q=3\) lanes \(3,3^2,3^3,3^4,3^5\), with periods 3, 9, 27, 81, 243 and the lower \(v_3(n)\) height strip. The lifted-copy audit reports 0 errors across 12 checked points. This directly visualizes exact nested divisibility. |
| `Screenshot 2026-09-07 063335.png` | Composite-support wave and prime-clock lanes near \(S=459{,}824{,}117\). The displayed prime-hole rule is \(H(n)=0\), with 2,406 required clocks known before the seed. The selected first gap is \(459{,}824{,}117\to459{,}824{,}137\), length 20. The raw middle statistic is small and positive in that run, but previous controlled tests show that no universal middle-rise law is established. |
| `Screenshot 2026-09-07 201848.png` | Seed-2, horizon-5,000 layer-birth view: 668 completed gaps, 11 run-relative record layers, 42 exact \(q^a\) thresholds crossed, and 126 twin-prime gaps. It distinguishes record factor depth from exact power-lane births. |
| `Screenshot 2026-09-08 110205.png` | Side-by-side 3D prime-power period banks for intervals 3–10,002 and 20,001–30,000. The screen reports all clocks, powers through \(a\le39\), 66,037 selected hits across both intervals, 31,989 marks in A and 34,048 in B, with no sampling. It supports the claim that the same exact lane laws persist at separated integer locations. |
| `Screenshot 2026-09-07 051424.png` | Clock spectrum for the frozen overlap field near \(S=1{,}327\), \(H=1{,}000\). Visible guides and measured peaks match fundamental clock fractions; the strongest nonconstant line is reported near \(f=0.4995\), the \(1/2\) line of clock 2. |
| `Screenshot 2026-09-07 053203.png` | A local 3D factor skeleton from prime boundary 13,267 to 13,291. Prime endpoints appear as zero-support boundary points, while composite integers carry stacked factor occurrences. |
| `Screenshot 2026-09-07 062122.png` | Front view of the factor skeleton around the 61,729 to 61,751 gap. The central integer-time spine and lateral factor ranks make the composite support between prime endpoints visible. |
| `zeta 1.png` | A 501-integer comparison beginning at 61,729 using 100,000 zero pairs. The screen reports \(\gamma_{100000}\approx74{,}920.827\), Pearson \(r=1.0000\) for the actual zero reconstruction, and \(r=0.1714\) for the scrambled-phase control. This is an exceptionally close cumulative residual match, but local event accuracy still requires first differences. |
| `zeta 2.png` | A 5,000-integer raw/common-units comparison beginning at 61,729 using 1,000 zeros. The screen reports 441 prime-power events, 3 higher powers, \(\gamma_{1000}\approx1{,}419.422\), Pearson \(r=0.9751\), and scrambled control \(r=0.4245\). |
| `zeta 3.png` | A 5,000-integer standardized-shape comparison beginning at 6,491 using 1,000 zeros. The screen reports 553 prime-power events, 9 higher powers, Pearson \(r=0.9804\), and scrambled control \(r=-0.3549\). Standardization removes amplitude information, so this is a shape result rather than an exact-units result. |

### 13.1 What is construction-driven and what is evidence

**Exact or construction-driven:** periodic \(q\)-clocks; nested \(q^a\)-lanes; the valuation sum; Fourier lines at clock frequencies; factor-skeleton placement rules; prime holes under a complete proper-divisor bank; the classical explicit-formula bridge from \(\Lambda\) and \(\psi\) to zeta zeros.

**Empirical evidence:** finite-window correlations; the visual similarity of separated lane banks; gap-clump shapes; any predictive value of higher-power context; any compression advantage; any scale at which a finite zero sum ranks individual prime events.

**Not established:** a new faster-than-sieve next-prime method; a universal rise toward the middle of gaps; equivalence between a zeta zero and an individual prime clock; proof of the Riemann hypothesis; historical novelty of the named project laws.

## 14. The central research hypothesis: compressing the phase field

The exact complete field already identifies primes, but it uses all required clocks. The scientifically important question is whether the field can be summarized without losing the location of its first zero.

### 14.1 Prime Phase Compression Hypothesis

A falsifiable working version is:

> There exists a predeclared, lower-cost summary of the seed phase vector and/or its local power geometry that ranks the next prime significantly better than equal-cost partial-sieve and randomized controls on disjoint, previously unseen intervals and across new orders of magnitude.

This is not yet a theorem. It becomes meaningful only after the feature budget, computational cost, training boundary, test seeds, and comparison baselines are fixed in advance.

### 14.2 Candidate route A: phase compression

Use a subset or transform of

\[
\Phi_{Q_N}(S)=\bigl(S\bmod q\bigr)_{q\le\sqrt N}
\]

or of the next-strike vector \((\tau_q(S))\). Candidate summaries include grouped residue histograms, spectral projections, low-dimensional embeddings, and carefully chosen subsets of clocks. The comparison must be against a partial sieve using the same clocks or an equal operation budget. A beautiful embedding is not enough; it must lower the rank of the true next prime out of sample.

### 14.3 Candidate route B: composite-neighborhood and power geometry

Higher powers cannot add new support beyond their base clocks, but the past and nearby power terrain may contain information about gap length or candidate ordering. Test features such as recent valuation profiles, lane densities, branch-envelope distances, and repeated completed-gap shapes.

The target interval's actual LPF, SPF, \(\Omega\), \(\omega\), complete factorization, future prime endpoint, and normalized target-gap coordinates are **post-reveal truth** and cannot be used as predictor inputs. Divisibility features explicitly computed from an allowed frozen clock bank are permitted, but their operation cost must be counted.

### 14.4 Candidate route C: zeta-assisted local ranking

The cumulative arithmetic residual is

\[
E_\psi(n)=\sum_{S<m\le n}\Lambda(m)-(n-S).
\]

Its exact first difference is

\[
E_\psi(n)-E_\psi(n-1)=\Lambda(n)-1.
\]

Therefore a finite zero reconstruction \(E_K\) implies the local estimator

\[
\boxed{
\widehat\Lambda_K(n)=1+E_K(n)-E_K(n-1).
}
\]

Normalize it by

\[
Z_K(n)=\frac{\widehat\Lambda_K(n)}{\log n}.
\]

For the exact \(\Lambda\)-signal, this normalized quantity is 1 at a prime, \(1/a\) at a higher prime power \(n=p^a\), and 0 at integers that are not prime powers. The finite approximation may overshoot, become negative, or oscillate; it is a score, not automatically a probability.

Let a deliberately incomplete clock bank stop at \(B<\sqrt N\):

\[
P_B(n)=\prod_{q\le B\atop q\ \mathrm{prime}}
\bigl(1-C_q(n)\bigr).
\]

Then the proposed hybrid score is

\[
\boxed{
R_{B,K}(n)=P_B(n)\,Z_K(n).
}
\]

This asks whether a partial sieve removes obvious composites and the finite zero sum ranks the survivors better than the same partial sieve alone. It is an experimental proposal, not a proved predictor. Its full cost includes calculating the zero sum, any smoothing, and all clock features.

### 14.5 Why cumulative zeta correlation is not yet prime prediction

The `zeta 1`–`zeta 3` plots compare cumulative residual curves. Such samples are strongly dependent, and broad low-frequency agreement can produce very high Pearson correlation without locating every individual \(\Lambda\)-jump. The stricter test is the first-difference sequence \(\widehat\Lambda_K(n)\), evaluated with raw, causal or unsmoothed data and held-out windows.

A useful local-resolution heuristic follows from

\[
x^{1/2+i\gamma}=\sqrt x\,e^{i\gamma\log x}.
\]

The phase rate in ordinary integer position is approximately \(\gamma/x\), giving wavelength

\[
\Delta x\approx\frac{2\pi x}{\gamma}.
\]

With \(\gamma_{\max}\approx74{,}920.827\), this is about 5.2 integers near \(x=62{,}000\), but about 38,600 integers near \(x=459{,}824{,}117\). This does not invalidate cumulative reconstruction; it warns that a zero table giving fine local detail at one scale will not automatically resolve individual primes at a vastly larger scale. The exact local engine remains the clock field unless the retained zero range grows substantially.

## 15. Frozen prediction protocol

Every serious prediction claim should use the following boundary.

### 15.1 Freeze

Before exposing target-region truth, fix:

- seed \(S\), horizon \(H\), and endpoint \(N\);
- clock cutoff \(B\) or full set \(Q_N\);
- zero count \(K\) and source/precision of zero ordinates;
- all feature definitions, smoothing rules, standardization, thresholds, and model parameters;
- test seeds and evaluation metrics;
- computation or operation budget for each baseline.

Allowed inputs are phases and known arithmetic information through \(S\), previously completed training intervals, and independently tabulated zeta-zero data.

### 15.2 Predict

Produce one locked output before revealing target truth:

- a single predicted next-prime offset;
- or a complete ranking of candidate offsets;
- or a calibrated score for every candidate in the horizon.

Centered smoothing is not allowed because it uses future neighbors. Standardization parameters must come from training data or a predeclared formula, not the target interval.

### 15.3 Reveal and audit

Only after the output is saved should the target region be factored and labeled. Then evaluate:

- exact next-prime accuracy;
- top-3, top-5, and top-10 coverage;
- rank of the true next prime among surviving candidates;
- number of candidates examined before the true next prime;
- false-positive and false-negative behavior for local \(\Lambda\)-events;
- runtime and counted arithmetic operations;
- performance as \(S\), \(H\), \(B\), and \(K\) change;
- stability on disjoint seeds and new orders of magnitude.

### 15.4 Required baselines and controls

At minimum compare:

1. full segmented sieve;
2. equal-clock or equal-operation partial sieve;
3. earliest survivor of the partial sieve;
4. partial sieve plus power/geometry features;
5. partial sieve plus \(Z_K\) or \(R_{B,K}\);
6. phase-scrambled zeta control;
7. shifted or permuted feature controls;
8. a no-training rule fixed from an earlier development set.

The essential ablation is

\[
\text{base clocks}
\quad\text{versus}\quad
\text{base clocks plus higher-power context}.
\]

If the higher-power context does not improve held-out ranking after equal-cost control, it remains a valuable explanatory geometry rather than a predictive variable. A negative result should be retained in the journal because it narrows the theory.

## 16. Data exports required for the first formal test

Screenshots preserve the ideas; raw primitive data are needed for equations and audits. Export the following with one row definition documented for each table.

### 16.1 Run metadata

- app/file version and checksum if available;
- export timestamp;
- \(S,H,N\);
- complete versus sampled mode;
- clock cutoff, power cutoff, zero count, smoothing, scaling, and all visible settings.

### 16.2 Seed clock-state table

One row per prime clock:

| Field | Meaning |
|---|---|
| `q` | prime clock |
| `seed_mod_q` | \(S\bmod q\) |
| `next_strike_offset` | \((-S)\bmod q\) |
| `first_strike_n` | first struck integer after \(S\) |
| `max_power_a` | largest retained \(a\) |
| `q_power_max` | \(q^{a}\) for that retained depth |

### 16.3 Candidate table

One row for every integer \(n=S+t\) in the horizon:

- `t`, `n`;
- proper base-clock hit count;
- lowest and/or first striking prime;
- survivor indicators for a predeclared sequence of cutoffs \(B\);
- nested power-hit height;
- candidate rank under each frozen score;
- post-reveal truth fields clearly separated.

### 16.4 Sparse power-hit ledger

One row for each actual hit:

\[
(n,q,a,q^a).
\]

This is preferable to a huge dense matrix and allows exact rebuilding of valuation heights and spectra.

### 16.5 Post-reveal truth table

Mark these columns as audit-only:

- prime/composite truth;
- SPF and LPF;
- \(\Omega\) and \(\omega\);
- full factorization;
- next-prime boundary and gap membership.

### 16.6 Zeta comparison table

For every integer sample export:

- \(n\), \(x=n+1/2\), and \(x_0\);
- \(\Lambda(n)\);
- exact \(E_\psi(n)\);
- \(E_K(n)\) for each requested \(K\);
- phase-scrambled control;
- first differences and, if the app can provide them, per-candidate \(\widehat\Lambda_K(n)\);
- \(K\), \(\gamma_{\max}\), smoothing mode, scaling mode, and zero-table provenance.

### 16.7 First canonical runs

1. **High-scale clock field:** \(S=459{,}824{,}117\), \(H=5{,}000\). Export the full clock-state table, candidate table, sparse power hits, and post-reveal truth.
2. **Zeta locality run:** \(S=61{,}729\), \(H=5{,}000\), raw and unsmoothed. Export once with \(K=1{,}000\) and once with \(K=100{,}000\), retaining exact common units and the phase control.

These two runs place the strongest high-scale clock evidence and the strongest finite-zero evidence on one auditable footing.

## 17. Current strongest formulation

The project can now be stated in two layers:

> **Exact theory.** Primes in a bounded future interval are exactly the zero set of a deterministic residue-phase field generated by the required earlier prime clocks. The same set of holes persists when nested prime-power lanes are added.

> **Research conjecture.** That high-dimensional phase field may have a smaller geometric or spectral summary capable of locating or ranking its next zero better than matched arithmetic baselines. The higher-power terrain and the local first difference of the zeta reconstruction are the leading candidate summaries.

The next evidence threshold is not another attractive plot. It is a locked, out-of-sample, equal-cost comparison using the exports in §16.


## 18. Wavefield laboratory converted into a research-export source — 2026-09-08

### 18.1 Retained visual base

The uploaded standalone application

`prime-composite-wavefield-lab_zeta_ LAB Final v1.html`

was judged to be a strong base for the next phase. It already contains the dense wave field, 3D factor skeleton, stacked side projection, single-prime power lanes, all-clock power sum, zeta reconstruction, numerical arithmetic test, arithmetic-to-complex transform, large-range sampler, aligned gap controls, layer births, and clock spectrum.

A modified file was created:

`prime-composite-wavefield-lab_RESEARCH-EXPORTS_v2.html`

SHA-256 at creation:

`be76961ae4d94d5f4d76321cba6309429d36d86331ebc69db190b649af12790c`

The change preserves the visual pages and adds a dedicated **Research exports** tab plus contextual CSV buttons on the relevant views.

### 18.2 Research-export information boundary

The new page distinguishes two data classes.

**Pre-reveal exports** may contain only:

- the fixed seed \(S\), horizon \(H\), and endpoint \(N\);
- a predeclared bank of prime clocks;
- the seed residues \(S\bmod q\), phase coordinates, and next-strike distances;
- future divisibility by \(q\) and \(q^a\) obtained by continuing those clocks;
- survivor order under that fixed bank.

**Post-reveal exports** contain:

- target prime/composite truth;
- full factorization, SPF, LPF, \(\Omega\), and \(\omega\);
- target complete-gap boundaries and normalized positions;
- factor skeletons and branch-owner measurements;
- performance scores calculated after the prediction is locked.

Important limitation: the browser lab still computes target factorizations internally for its descriptive pictures. The export split is therefore a disciplined schema boundary, not yet a physically isolated two-process Freeze → Predict → Reveal engine. The later standalone script should enforce that boundary at the file/process level.

### 18.3 Configurable pre-reveal clock banks

The Research exports page supports:

1. a prime cutoff \(q\le B\);
2. the first \(K\) required clocks;
3. the complete bank \(q\le\sqrt N\).

It also supports an automatic or fixed maximum power exponent and either future-only rows \(S+1,\ldots,N\) or rows including the seed.

The page immediately reports:

- selected clock count and largest selected clock;
- partial versus complete bank status;
- number of clock survivors;
- revealed prime count and false-survivor count as an audit-only summary.

This creates the central experimental comparison:

\[
\text{partial base clocks}
\quad\text{versus}\quad
\text{partial base clocks plus }q^a\text{ context}
\quad\text{versus}\quad
\text{complete bounded bank}.
\]

### 18.4 New core files

#### Clock state at seed

One row per selected prime \(q\), including:

- \(S\bmod q\);
- normalized phase, phase angle, sine, and cosine;
- previous strike and first strike after \(S\);
- distance to that next strike;
- \(q^2\);
- active power lanes at the seed and through \(N\);
- next prime-power threshold.

The identity

\[
(S+t)\bmod q=((S\bmod q)+t)\bmod q
\]

means this seed table plus the offset \(t\) reconstructs the complete future phase vector for the selected bank.

#### Candidate pre-reveal table

One row per candidate, containing:

- \(n\) and \(t=n-S\);
- selected base-clock hit count and hitting primes;
- selected-bank survivor indicator and survivor rank;
- total selected power hits;
- higher-power excess above the base strikes;
- maximum selected valuation;
- deepest selected power period;
- compact valuation signature.

No prime label, LPF, SPF, \(\Omega\), or target factorization appears in this file.

#### Sparse power-hit ledger

One row for every selected event

\[
(n,q,a,q^a),
\]

with its period, frequency, full \(v_q(n)\), and higher-power flag.

#### Post-reveal truth table

One row per integer with:

- primality and full factorization;
- SPF, LPF, canonical LPF cofactor, \(\Omega\), and \(\omega\);
- log support and normalized support;
- full required-clock hits;
- prime-power base, exponent, and \(\Lambda(n)\);
- strict previous and next primes;
- complete-gap membership, offset, and normalized phase where defined.

#### Complete-gap features

One row per completed gap with:

- length and twin status;
- maximum \(\Omega\), deepest integer, mean excess, log-support energy, maximum \(v_2\), and maximum \(v_3\);
- raw middle-minus-outer tests under several metrics;
- largest LPF owner \(p_{\max}\), selected landing \(b^*\), multiplier \(k\), and distance \(d\);
- selected-bank base/power strike totals and partial-bank survivors;
- exact selected-clock counts and prime-power events.

#### Factor-occurrence coordinates

One row per prime-factor occurrence, retaining:

- integer time and normalized time;
- occurrence rank from the largest factor;
- LPF/support role;
- alternating side slot;
- exact prime factor and repeated-factor occurrence number;
- log height and normalized log height.

### 18.5 Contextual exports added to the visual pages

The following pages now export their displayed mathematical objects directly:

- Wave field — current scope, metric, raw signal, centered display signal, and integer truth.
- 3D factor skeleton — selected-gap factor occurrences.
- 2D stacked side — factor occurrences for the selected overlaid gap group.
- Prime-power waves — every displayed \(q^a\) lane in long format.
- All-clock power sum — displayed valuation columns and exact power signatures.
- Large-range sampler — every retained sample with block and rotating-offset metadata.
- Gap clumps + control — each aligned 61-point observed curve, group summaries, and parity-control mean.
- Layer births + repeats — gap features, record events, selected-clock counts, and thresholds.
- Clock spectrum — all Fourier bins, powers, periods, nearest source classifications, and resolution flags.

The earlier zeta, arithmetic-block, and complex-plane exports remain.

### 18.6 Local zeta-event export

The zeta CSV was expanded beyond cumulative correlation. For integer \(n\), it now exports

\[
\Delta E_\psi(n)=E_\psi(n)-E_\psi(n-1),
\]

\[
\Delta E_K(n)=E_K(n)-E_K(n-1),
\]

and

\[
\widehat\Lambda_K(n)=1+\Delta E_K(n).
\]

The phase-control analogue and the normalized scores

\[
\frac{\widehat\Lambda_K(n)}{\log n}
\]

are also exported. Additional fields record \(K\), \(\gamma_{\max}\), smoothing, scale, and the rough finest ordinary-integer wavelength

\[
\Delta x_{\min}(n)\approx\frac{2\pi(n+1/2)}{\gamma_{\max}}.
\]

This makes the correct next zeta test possible: local prime-power event recovery, not only cumulative-curve agreement.

### 18.7 Corrections incorporated

- The all-clock power routine in this uploaded version already applies the proper-divisor guard \(q<n\); that behavior was retained.
- The Wave field now explicitly warns that \(\omega(n)-1\) also vanishes at composite prime powers.
- The yellow layer-ring caption now reports actual record increases among completed gaps. It no longer assumes automatic equality with the global powers-of-two envelope when the newest power of two may lie in a trailing incomplete gap.

### 18.8 Regression and browser checks

For the default run \(S=61{,}729,H=500\):

- the complete bank has 53 clocks and produces 43 survivors, exactly the 43 future primes;
- the cutoff \(q\le31\) has 11 clocks and produces 72 survivors: 43 revealed primes and 29 false survivors;
- the new full-bank candidate file matches all 500 rows of the earlier frontier CSV in strike count, striking-prime list, and coverage status;
- the enhanced zeta export satisfies \(1+\Delta E_\psi(n)=\Lambda(n)\) on every non-seed row;
- all new contextual and core download buttons were exercised in a headless Chromium run with zero JavaScript page errors.

For \(S=459{,}824{,}117,H=500\), the complete bank contains 2,406 clocks and produces 25 exact prime holes in the tested interval. The cutoff \(q\le31\) leaves 72 candidates, of which 47 are revealed composites. This confirms that the same export machinery operates at the high-scale screenshot seed.

### 18.9 Immediate experimental use

The preferred high-scale run remains

\[
S=459{,}824{,}117,\qquad H=5{,}000.
\]

Before running it, predeclare a cutoff sequence such as

\[
B\in\{31,97,251,997\}
\]

plus matched first-\(K\) banks and the full bank. Export every run rather than selecting only the best result afterward.

The preferred zeta-locality run remains

\[
S=61{,}729,\qquad H=5{,}000,
\]

with raw, exact-common-unit exports at \(K=1{,}000\) and \(K=100{,}000\).

These files are now sufficient to build the standalone `prime_phase_experiment.py` pipeline and to compare base clocks, higher-power context, geometry/history, and local zeta scores under one fixed information boundary.

## 19. High-scale cutoff audit: q <= 97 at S = 459,824,117

This section records the first completed intermediate-bank run from the Research Exports v2 page. The source package is `prime-research-package-S459824117-H5000-cutoff (1).json`, generated with the seed excluded from the candidate table.

### 19.1 Frozen run definition

\[
S=459{,}824{,}117,\qquad H=5{,}000,\qquad N=459{,}829{,}117.
\]

The selected pre-reveal bank was

\[
Q_{97}=\{q\text{ prime}:q\le97\},
\]

containing 25 clocks. Candidate scope was future only, so offsets were exactly \(t=1,\ldots,5000\). The bounded exact bank would require all 2,406 primes through 21,433, so this was deliberately an incomplete partial sieve.

### 19.2 Export audit

The package contained:

- 25 seed clock-state rows;
- 5,000 pre-reveal candidate rows;
- 12,868 sparse \(q^a\)-hit rows;
- 5,000 post-reveal truth rows;
- 256 completed consecutive-prime gaps;
- 20,272 factor-occurrence rows.

Independent recomputation found zero discrepancies in seed residues, next-strike distances, base hit counts, striking-prime lists, survivor flags, selected power-hit totals, maximum selected valuations, or primality labels.

### 19.3 Compression achieved by the partial bank

The interval contains 256 primes and 4,744 composites. With clocks through 97:

- 600 candidates survived;
- all 256 primes survived;
- 344 composites survived falsely;
- 4,400 of 4,744 composites were eliminated;
- composite elimination was 92.7487%;
- prime precision among survivors was 42.6667%.

For comparison:

| Bank | Clocks | Survivors | False survivors | Composite elimination | Prime fraction among survivors |
|---|---:|---:|---:|---:|---:|
| first 10 clocks, through 29 | 10 | 788 | 532 | 88.7858% | 32.4873% |
| cutoff through 31 | 11 | 762 | 506 | 89.3339% | 33.5958% |
| cutoff through 97 | 25 | 600 | 344 | 92.7487% | 42.6667% |
| full bounded bank through 21,433 | 2,406 | 256 | 0 | 100% | 100% |

Moving from 31 to 97 removed 162 candidates, all composites. Their smallest prime factors were precisely among the newly added clocks 37 through 97. No prime was removed.

### 19.4 Next-prime candidate ranks across all 256 gaps

For every completed prime gap, rank the partial-bank survivors strictly after the left prime. Under the \(q\le97\) bank, the true right-hand prime had these ranks:

- rank 1 in 109 of 256 gaps: 42.5781%;
- rank at most 2 in 174 gaps: 67.9688%;
- rank at most 3 in 212 gaps: 82.8125%;
- rank at most 5 in 240 gaps: 93.7500%;
- rank at most 10 in all 256 gaps.

The mean number of false survivors before the next prime was 1.34375; the maximum was 9. The corresponding \(q\le31\) values were 34.7656% at rank 1, 71.8750% by rank 3, 87.5000% by rank 5, and a maximum rank of 14.

The \(q\le97\) rank distribution is very close to the geometric null obtained by treating each survivor as prime with the overall survivor precision 256/600. In particular, the observed rank-1 rate 42.5781% is nearly identical to the overall survivor prime fraction 42.6667%. Thus the first survivor does not yet show an extra boundary advantage beyond the ordinary partial-sieve density.

### 19.5 The opening gap and its exact obstruction

The first two \(q\le97\) survivors were

\[
459{,}824{,}129=3449\times133321
\]

and

\[
459{,}824{,}137,
\]

where the second is the next prime. Therefore the next prime has local survivor rank 2. Any ordinary cutoff below 3,449 must leave the earlier composite untouched, because its smallest prime factor is 3,449. Cutoffs 251 and 997 can reduce later false survivors but cannot make this particular next prime rank 1. A cutoff including clock 3,449 removes the obstruction.

### 19.6 Candidate-local higher powers cannot rank the survivors

Every one of the 600 \(q\le97\) survivors has

\[
\text{selected power-hit count}=0,
\]

\[
\text{higher-power extra count}=0,
\]

and maximum selected valuation 0. This is forced: if \(q^a\mid n\), then \(q\mid n\). Once a candidate survives every base clock in the selected bank, it also survives every higher-power lane belonging to that bank.

This resolves an important design question. Higher powers at the candidate itself cannot distinguish a true prime from a false survivor of the same base bank. Any additional power-derived predictor must use the surrounding continued field, phase distances, or temporal neighborhood—not the candidate's own selected valuation height.

### 19.7 Structure of the remaining false survivors

All 344 false survivors have smallest prime factor at least 101. Their factor depths are:

- 291 with \(\Omega=2\), about 84.6%;
- 53 with \(\Omega=3\), about 15.4%;
- no composite prime powers;
- only two with repeated prime factors.

The residual problem after the small clocks is therefore dominated by products of two larger primes. To the retained clock bank, these numbers look exactly like primes at the candidate location: every selected base and power lane is zero.

### 19.8 Wheel-density control

The observed survivor density is

\[
600/5000=0.120000.
\]

The ordinary modular-wheel prediction is

\[
\prod_{q\le97}\left(1-\frac1q\right)\approx0.120317.
\]

The close match is a strong implementation control. It shows that the visible survivor set has the expected density of integers coprime to the primorial through 97. It is not, by itself, evidence of an extra predictive law.

### 19.9 Larger cutoffs can be reconstructed from this audited truth table

Because the post-reveal table contains the SPF of every candidate, the descriptive effect of any larger cutoff in this same already-revealed interval can be reconstructed exactly. Selected examples are:

| Cutoff B | Survivors | False survivors | Survivor prime fraction | Rank-1 next-prime rate | Maximum next-prime rank |
|---:|---:|---:|---:|---:|---:|
| 97 | 600 | 344 | 42.67% | 42.58% | 10 |
| 251 | 497 | 241 | 51.51% | 51.56% | 9 |
| 997 | 404 | 148 | 63.37% | 63.28% | 6 |
| 3,449 | 343 | 87 | 74.64% | 76.95% | 5 |
| 10,007 | 291 | 35 | 87.97% | 88.67% | 3 |
| 21,433 | 256 | 0 | 100% | 100% | 1 |

These are post-reveal descriptive reconstructions, not independent prediction trials. They mean that additional manual exports at 251 and 997 are unnecessary for this particular interval. Future blind intervals must still declare the bank before truth is revealed.

### 19.10 Consequence for the next experiment

The exact base field is working and the partial bank is a powerful compressor, but the retained candidate-local power data contain no ranking information. The next genuine experiment should therefore test a frozen ranking function on the partial-sieve survivors using only legal information such as:

- the complete nonzero residue/phase vector \((n\bmod q)_{q\le97}\);
- distance to previous and next selected-clock strikes;
- a predeclared neighborhood of the continued base/power field;
- past-only gap and branch summaries;
- optionally, a separately evaluated local zeta score.

The baseline is not merely 'first survivor.' It is the geometric/wheel null implied by the observed survivor precision. A useful theory must improve held-out rank-1, top-k, or candidate-count performance beyond that baseline on a new interval whose truth was not used to choose the formula.

---

### Recorder update ledger — 2026-09-08

- Canonical recorder refreshed after the \(q\le97\) high-scale audit.
- Source run: `prime-research-package-S459824117-H5000-cutoff (1).json`.
- Full numerical audit and conclusions are retained in §19.
- Next planned work: construct and freeze a phase/neighborhood ranking test, then evaluate it on a new unrevealed interval.

## 20. High-zero finite zeta reconstruction audit at S = 61,729

This section records the first direct higher-cutoff comparison using the external `zeros6.gz` table. Both exports used raw, unsmoothed values and exact common vertical units. The relevant source files are:

- `prime-zeta-comparison-S61729-H500-K1000000.csv`;
- `prime-zeta-comparison-S61729-H5000-K500000.csv`.

The second file contains the same first 500 future integers as the first file, so the first 500 rows provide a controlled apples-to-apples comparison between 500,000 and 1,000,000 retained positive zero pairs.

### 20.1 Run definitions

**One-million-zero run**

\[
S=61{,}729,\qquad H=500,\qquad K=1{,}000{,}000.
\]

The highest retained ordinate was approximately

\[
\gamma_{\max}=600{,}269.677012445.
\]

The rough finest ordinary-integer wavelength ranged from about 0.646 to 0.651 integer across the window. There were 43 future prime-power events, all ordinary primes, after the seed.

**Five-hundred-thousand-zero run**

\[
S=61{,}729,\qquad H=5{,}000,\qquad K=500{,}000.
\]

The highest retained ordinate was approximately

\[
\gamma_{\max}=319{,}387.21930923.
\]

The rough finest wavelength ranged from about 1.214 to 1.313 integers. There were 441 future prime-power events after the seed: 438 ordinary primes and three higher prime powers.

### 20.2 Direct same-window effect of doubling K

Restrict the 500,000-zero file to the same 500 future integers as the 1,000,000-zero run. The exact arithmetic sequence is identical row for row, so the only material difference is the retained zero cutoff.

| Metric on the same 500 future integers | 500,000 zeros | 1,000,000 zeros |
|---|---:|---:|
| Cumulative Pearson correlation | 0.9997606643 | 0.9999794462 |
| Cumulative mean absolute error | 0.80476755 | 0.09799535 |
| Cumulative root-mean-square error | 0.87374657 | 0.12144213 |
| Maximum cumulative absolute error | 1.93237563 | 0.38009425 |
| Local one-step correlation | 0.9938054182 | 0.9978943386 |
| Local one-step mean absolute error | 0.38184192 | 0.16156152 |
| Local one-step root-mean-square error | 0.60918317 | 0.20071299 |

Thus doubling the retained zeros reduced cumulative mean absolute error by about 87.8%, cumulative RMS error by about 86.1%, local mean absolute error by about 57.7%, and local RMS error by about 67.0% on this fixed window. This is direct numerical evidence that the extra zeros add real resolution and improve amplitude accuracy, rather than merely making the same plot look denser.

Both cutoffs already separated all 43 future ordinary primes from all composites in this short window using the normalized local score. The one-million run therefore improves numerical reconstruction substantially even though the binary prime/nonprime ranking was already saturated at 500,000 zeros here.

For the one-million run, the normalized local scores were:

- ordinary primes: approximately 0.9673 through 1.0149;
- ordinary composites: approximately -0.0563 through 0.0569.

This is a large clean observed gap, but it remains an in-sample result until the scoring rule and threshold are frozen and applied to an untouched interval.

### 20.3 Five-thousand-integer result with 500,000 zeros

Across the full 5,000-future-integer window:

- cumulative correlation between exact and zero reconstruction: 0.9999594191;
- cumulative phase-control correlation: 0.4192071873;
- local one-step correlation: 0.9931888758;
- local phase-control correlation: -0.0126504471;
- cumulative mean absolute error: 0.80375043;
- local mean absolute error: 0.38853009.

The displayed cumulative correlation rounds to `1.0000`, but the curves are not numerically identical. The cumulative difference has a mean of about 0.7988 units, showing why correlation alone is insufficient: it is insensitive to an almost constant vertical displacement. The local phase-control correlation is the more relevant control for event recovery and is essentially zero.

For ordinary-prime identification, the normalized local score had these observed ranges:

- 438 ordinary primes: 1.08777 through 1.17427;
- 4,559 ordinary non-event composites: -0.19657 through 0.07563;
- higher prime powers: separate fractional-scale values.

The three higher powers were:

| Integer | Exact type | Normalized reconstructed score |
|---:|---|---:|
| 63,001 | \(251^2\) | 0.57410 |
| 65,536 | \(2^{16}\) | 0.01918 |
| 66,049 | \(257^2\) | 0.58162 |

A threshold of 0.75 would retrospectively classify all 438 ordinary primes and no nonprimes in this observed window. The actual-score prime AUC is 1.000, while the phase-scrambled score has AUC about 0.488, consistent with chance ordering. This is strong finite-window evidence, but the threshold was inspected after the result and therefore must not yet be reported as a validated predictor.

The high-exponent event \(2^{16}\) is not recovered with its ideal normalized amplitude, showing that even when ordinary-prime separation is clean, recovery of every prime-power exponent is not exact.

### 20.4 Interpretation and status

These runs support four specific conclusions:

1. **Increasing the zero cutoff improves the finite explicit-formula reconstruction directly and substantially on a fixed integer window.**
2. **At the scale near 61,729, the retained spectra reach approximately integer or sub-integer resolution and recover ordinary-prime jumps extremely cleanly in the tested windows.**
3. **A displayed cumulative correlation of 1.0000 is rounded shape agreement, not exact equality. Error magnitude and one-step recovery must always be reported beside correlation.**
4. **Correct zero phases matter.** The phase-scrambled control does not recover the local prime signal.

This does not constitute a new proof of the classical zeta/prime-power relation; the explicit formula already supplies that deterministic connection. The potentially new experimental question is whether a fixed finite zero sum can be used as a practical, frozen ranking signal for partial-sieve survivors over new intervals and increasing numerical scales.

### 20.5 Next required test

Before inspecting another target interval, lock a local ordinary-prime rule. A natural first rule, motivated by the observed separation but not yet validated, is:

\[
\widehat{\Lambda}_K(n)/\log n > 0.75.
\]

Then choose a new prime seed and horizon in advance, retain raw unsmoothed exact-unit output, compute the finite zero reconstruction, and reveal the prime labels only after the prediction file is locked. Report exact sensitivity, false positives, candidate rank, AUC, local error, and the phase-scrambled control. Repeat at multiple scales without changing the threshold in response to test outcomes.

---

### Recorder update ledger — high-zero audit

- Added the direct 500,000-versus-1,000,000 zero comparison at a fixed window.
- Added cumulative and local error metrics, not correlation alone.
- Recorded perfect observed ordinary-prime separation in the two current windows as an in-sample finding only.
- Recorded the three higher prime-power events and the imperfect recovery of \(2^{16}\).
- Next planned work: pre-register one local score threshold and test it on untouched windows.

## 21. First prospectively frozen finite-zeta threshold validation

### 21.1 Why this run is different

Before this target interval was generated, the following local classification rule was explicitly fixed from the prior exploratory windows:

\[
\boxed{\widehat{\Lambda}_{K}(n)/\log n>0.75\quad\Longrightarrow\quad\text{predict ordinary prime}.}
\]

The new test then used:

- seed \(S=70{,}001\), which was treated as the already-known starting prime;
- future horizon \(H=500\), covering \(70{,}002\) through \(70{,}501\);
- the first \(K=1{,}000{,}000\) positive zeta-zero ordinates from the loaded `zeros6.gz` table;
- raw, unsmoothed comparison values;
- exact common vertical units;
- the same phase-scrambled control already built into the lab.

The rule and threshold were not changed after inspecting this interval. This makes the run the project's first prospectively frozen numerical validation of the local finite-zero score. It is not yet a physically sealed blind experiment because the visualization program computes arithmetic truth in the same process for scoring, but the decision rule itself was genuinely predeclared in the conversation before the file was produced.

### 21.2 Independent audit of the export

The export contains 501 integer samples: the seed plus exactly 500 future integers. Independent checks found:

- 50 ordinary primes among the 500 future integers;
- 450 composites;
- no higher prime powers in the future interval;
- zero mismatches in the CSV prime labels under an independent primality test;
- the exact identity \(1+\Delta E_\psi(n)=\Lambda(n)\) holds to floating-point tolerance on every future row.

The displayed cumulative Pearson value `0.9999` corresponds to the fuller value:

\[
0.9999036051.
\]

The cumulative phase-scrambled control correlation is:

\[
0.1756714748.
\]

The more relevant one-integer local correlations are:

\[
\operatorname{corr}(\Lambda,\widehat\Lambda_K)=0.9941352922,
\]

and for the phase-scrambled control:

\[
0.0477347198.
\]

Thus the correct phases recover the local arithmetic events strongly, while the control is near chance at the local level.

### 21.3 Frozen threshold result

Applying the already-fixed threshold of 0.75 to all 500 future integers produced:

| Outcome | Count |
|---|---:|
| True primes correctly predicted | 50 |
| Composite false positives | 0 |
| Composites correctly rejected | 450 |
| Missed primes | 0 |

Therefore, on this untouched interval:

- sensitivity / recall = 100%;
- specificity = 100%;
- precision = 100%;
- total accuracy = 100%;
- rank AUC = 1.000.

The phase-scrambled control, using the same 0.75 threshold, found no true primes, falsely selected the composite 70,048, and missed all 50 primes. Its rank AUC was approximately 0.531, consistent with chance ordering.

### 21.4 Separation margin

The finite-zero normalized scores occupied two widely separated ranges:

- all 50 primes: 1.06248121 through 1.13630609;
- all 450 composites: -0.14274373 through 0.08328653.

The lowest prime score occurred at \(70{,}117\). The highest composite score occurred at \(70{,}239\). The observed gap between the largest composite and smallest prime was:

\[
1.06248121-0.08328653\approx0.97919468.
\]

Therefore the predeclared threshold 0.75 did not succeed by sitting on a fragile boundary; it lay well inside a large empty interval between the two observed classes. Retrospectively, any strict threshold from approximately 0.0833 up to but not including 1.0625 would separate this particular sample, but only 0.75 has prospective status because it was fixed in advance.

### 21.5 Error measurements

Over the 500 future integers:

| Measurement | Value |
|---|---:|
| Cumulative correlation | 0.99990361 |
| Cumulative mean absolute error | 0.70249113 |
| Cumulative RMS error | 0.75148182 |
| Maximum cumulative absolute error | 1.53540907 |
| Local one-step correlation | 0.99413529 |
| Local mean absolute error | 0.38251091 |
| Local RMS error | 0.52650623 |
| Maximum local absolute error | 1.59341570 |

The cumulative reconstruction has a nearly constant displacement of about 0.70 units over this window, yet the local event score still separates the classes perfectly. This again shows why cumulative correlation or cumulative vertical agreement is not the decisive prediction measure; the one-step normalized jump is.

### 21.6 What this establishes—and what it does not

This run establishes a genuine empirical statement:

> With the rule and threshold fixed before inspection, the first one million positive zero ordinates classified every ordinary prime and every composite correctly in the untouched interval 70,002–70,501.

This is stronger than the earlier in-sample observations. It is the first actual out-of-sample success of the finite-zeta threshold program.

It does **not** yet establish:

- that the threshold works for all integers;
- that it remains valid as the numerical scale grows;
- that one million zeros suffice at larger scales;
- that the method is computationally competitive with sieving;
- that higher prime powers can never cross the threshold, because this particular target interval contained none;
- a new proof of the explicit formula, primality, or the Riemann hypothesis.

The finite sum uses much more work than an ordinary sieve in this form. Its importance is structural: a fixed spectral truncation produced a clean local prime indicator on a genuinely new interval.

### 21.7 Required replication sequence

The next evidence should preserve the same score definition and threshold without tuning. The replication program should include:

1. additional disjoint 500-integer windows near the same scale;
2. a deliberately chosen window containing one or more higher prime powers, to test false-positive resistance;
3. progressively larger seeds, while recording the highest zero ordinate and the rough finest integer wavelength;
4. failures as well as successes, with no threshold adjustment between runs;
5. eventual process separation: write the score predictions first, hash or save them, then calculate or import truth in a separate reveal step.

---

### Recorder update ledger — first frozen zeta validation

- Added the predeclared 0.75 threshold and exact run configuration.
- Recorded the 50/50 prime recovery and 450/450 composite rejection.
- Added cumulative and local correlations, absolute errors, score ranges, and the phase-scrambled control.
- Marked the result as the first genuine out-of-sample numerical success, while retaining the limits on generality, efficiency, prime-power coverage, and process sealing.
- Next planned work: replicate without changing the rule.


## 22. Five-thousand-integer frozen zeta stress validation with prime squares

### 22.1 Predeclared rule and purpose

The ordinary-prime classification rule remained fixed from the earlier exploratory work and the first 500-integer prospective validation:

\[
\boxed{\widehat{\Lambda}_{K}(n)/\log n>0.75\quad\Longrightarrow\quad\text{predict ordinary prime}.}
\]

No threshold or score definition was changed for this run. The interval was deliberately chosen as a stress test because it contains at least one known prime square; the user then expanded the requested horizon from 500 to 5,000, making this a ten-times-longer test than the first prospective validation.

Run settings:

- seed \(S=72{,}019\), treated as the already-known starting prime;
- future horizon \(H=5{,}000\), covering \(72{,}020\) through \(77{,}019\);
- first \(K=1{,}000{,}000\) positive zeta-zero ordinates loaded from `zeros6.gz`;
- highest retained ordinate \(\gamma_{\max}=600{,}269.677012445\);
- raw, unsmoothed comparison values;
- exact common vertical units;
- the same phase-scrambled control as in prior runs.

Source export:

- `prime-zeta-comparison-S72019-H5000-K1000000.csv`
- SHA-256: `7d6660e47b5ffd7697db81e7f37a4258c4b38850208e188c8b93f6c9d9c50a2f`

### 22.2 Independent arithmetic audit

The export contains 5,001 rows: the seed plus exactly 5,000 future integers. An independent sieve and prime-power audit found:

- 440 ordinary primes;
- 4,560 nonprimes;
- 443 total von Mangoldt events;
- exactly three higher prime powers, all prime squares;
- zero mismatches in the CSV prime labels;
- zero mismatches in the exported exact \(\Lambda(n)\) values.

The three nonprime prime-power events are:

| Integer | Exact form | Ideal normalized event height |
|---:|---|---:|
| 72,361 | \(269^2\) | \(1/2\) |
| 73,441 | \(271^2\) | \(1/2\) |
| 76,729 | \(277^2\) | \(1/2\) |

### 22.3 Frozen threshold result

Applying the already-fixed threshold of 0.75 to every future integer produced:

| Outcome | Count |
|---|---:|
| True primes correctly predicted | 440 |
| Composite false positives | 0 |
| Composites correctly rejected | 4,560 |
| Missed primes | 0 |

Therefore, for this 5,000-integer stress window:

- sensitivity / recall = 100%;
- specificity = 100%;
- precision = 100%;
- total accuracy = 100%;
- ordinary-prime rank AUC = 1.000.

The same frozen threshold also correctly rejected all three prime squares.

Across this run and the earlier disjoint 500-integer validation near 70,001, the predeclared rule has now classified 5,500 future integers with zero errors: 490 ordinary primes and 5,010 nonprimes. This is still a finite same-scale empirical result, not a universal theorem.

### 22.4 Separation margin and prime-square behavior

The normalized local finite-zero score ranges were:

- all 440 ordinary primes: 1.0730933194 through 1.1566521226;
- all 4,560 nonprimes: -0.1737899452 through 0.5722191401.

The observed gap between the lowest prime and highest nonprime was:

\[
1.0730933194-0.5722191401\approx0.5008741793.
\]

The fixed threshold 0.75 therefore had margins of:

- approximately 0.3231 below the weakest observed prime;
- approximately 0.1778 above the strongest observed nonprime.

The three prime-square scores were:

| Integer | Reconstructed normalized score |
|---:|---:|
| 72,361 | 0.5570876190 |
| 73,441 | 0.5516854244 |
| 76,729 | 0.5722191401 |

These values are above the ordinary composite background but remain well below the ordinary-prime band and below the frozen 0.75 threshold. The highest nonprime score in the entire run is the square \(277^2=76{,}729\). The highest score among non-event composites is only about 0.06434.

This is important evidence that the normalization is separating event multiplicity in the intended direction: ordinary primes cluster near one, while prime squares cluster near one-half.

### 22.5 Reconstruction and control statistics

For the 5,000 future integers:

| Measurement | Correct zero phases | Phase-scrambled control |
|---|---:|---:|
| Cumulative Pearson correlation | 0.9999003738 | 0.2206955059 |
| Local normalized-event correlation | 0.9940356309 | 0.0105482243 |

Additional correct-phase error measurements:

| Measurement | Value |
|---|---:|
| Cumulative mean absolute error | 0.5548189357 |
| Cumulative RMS error | 0.6127959592 |
| Maximum cumulative absolute error | 1.5570939673 |
| Local normalized mean absolute error | 0.0320776375 |
| Local normalized RMS error | 0.0509988117 |
| Maximum local normalized absolute error | 0.1737899452 |

Using the same 0.75 threshold on the phase-scrambled control produced only 4 true-prime detections, 32 composite false positives, and 436 missed primes. Thus the successful classification is not reproduced by retaining the same frequencies and amplitudes while destroying the actual zero phases.

The displayed value should be called Pearson \(r\), not a p-value. The full cumulative value is approximately \(r=0.9999004\); its significance here is descriptive curve agreement. A naive p-value would be inappropriate because successive cumulative samples are strongly autocorrelated.

### 22.6 Resolution scale

Across the interval, the reported rough finest wavelength from the retained zero height rises only from about 0.7539 to 0.8062 integer units. Thus the one-million-zero truncation remains in a sub-integer-resolution regime throughout this particular numerical window, consistent with the strong local event recovery.

This is a resolution guide rather than a proof that a fixed cutoff must succeed whenever the wavelength is below one. The actual finite explicit-formula error still depends on truncation, correction terms, phase interaction, and numerical implementation.

### 22.7 What this establishes—and what it does not

This run establishes the following empirical statement:

> With the ordinary-prime score and threshold fixed in advance, the first one million positive zeta-zero ordinates classified all 440 ordinary primes and all 4,560 nonprimes correctly in the 5,000-integer interval 72,020–77,019, while also rejecting three prime squares whose scores remained near one-half.

This is a materially stronger validation than the earlier 500-integer run because it is ten times longer and directly exercises the chief prime-power false-positive concern.

It does **not** yet establish:

- universal correctness of the 0.75 rule;
- correctness outside the current numerical scale;
- that one million zeros remain sufficient once the rough wavelength exceeds one integer;
- computational superiority over an ordinary sieve or primality test;
- a new proof of the explicit formula, the prime number theorem, or the Riemann hypothesis;
- complete recovery of every higher prime-power exponent at all scales.

The zero sum is a finite spectral reconstruction of the classical explicit-formula signal. Its present importance is that a fixed truncation and fixed local rule have now passed two disjoint prospective windows, including a deliberately selected prime-square stress interval.

### 22.8 Next evidence standard

Do not tune the 0.75 threshold in response to future outcomes. The next program should include:

1. multiple disjoint, predeclared windows rather than one-at-a-time favorable selection;
2. a scale ladder that crosses the approximate one-integer-resolution boundary for the current \(\gamma_{\max}\);
3. retention of every failure and near miss;
4. separate reporting of ordinary primes, prime squares, higher prime powers, and non-event composites;
5. a physically separated prediction file written before truth generation or import;
6. direct comparison with ordinary sieving cost and with a partial-clock candidate filter.

---

### Recorder update ledger — 5,000-point zeta stress validation

- Added the ten-times-longer frozen threshold test at \(S=72{,}019\), \(H=5{,}000\).
- Recorded perfect classification of 440 primes and 4,560 nonprimes.
- Recorded successful rejection of \(269^2\), \(271^2\), and \(277^2\).
- Added exact score ranges, threshold margins, cumulative/local errors, resolution scale, and phase-control results.
- Clarified that the displayed statistic is Pearson \(r\), not a p-value.
- Next planned work: predeclare a multi-window scale ladder without threshold tuning.

---

## 23. Prime-Power Coordinate Atlas and Gap-State Lab — first build

### 23.1 Why this laboratory was created

After the clock-field and finite-zero tests, the project deliberately paused further prediction tuning in order to build a clearer structural language. The immediate goal is not to force one final forecasting pipeline. It is to create a permanent atlas in which every prime-power period has one fixed address, then step through the integers and observe how all of those states evolve.

The design preserves the user's important methodological point: completed gap skeletons, branch shapes, and other revealed structures may later suggest reorganizations that reduce or simplify computation. They should not be discarded merely because they are post-reveal. Instead, the lab labels them clearly and allows them to be turned off when a causal-only view is needed.

### 23.2 Permanent atlas coordinate

Let the ordered primes be

\[
q_1=2,\ q_2=3,\ q_3=5,\ldots.
\]

The permanent atlas cell is

\[
\boxed{A(i,a)\longleftrightarrow q_i^a.}
\]

Here:

- \(i\) is the prime index of \(q_i\);
- \(a\) is the exponent;
- the lane period is \(m=q_i^a\);
- the lane frequency is \(1/m\) strikes per integer;
- the lane's first strike, or birth, is at integer \(m\).

The address never changes. For example,

\[
A(2,2)=3^2=9
\]

for every integer and every run.

This notation intentionally avoids using \(j\), because the existing factor skeleton already uses an occurrence rank / lateral slot. Atlas prime index and skeleton occurrence slot are separate coordinates.

### 23.3 Moving phase state

For current integer \(n\) and cell period \(m=q_i^a\), define

\[
r_{i,a}(n)=n\bmod m,
\qquad
\phi_{i,a}(n)=\frac{r_{i,a}(n)}{m}.
\]

The previous strike is \(n-r_{i,a}(n)\). The next strike after the current integer is:

- \(n+m\) when the cell strikes now;
- otherwise \(n+[m-r_{i,a}(n)]\).

In the visual atlas, every cell contains a one-cycle horizontal track. A dot marks \(\phi_{i,a}(n)\). Advancing the integer by one moves every dot by one step in its own period. Thus the \(q=2\) base lane alternates rapidly, while a large prime-power lane moves only a small fraction of its cycle at each step.

This is the first concrete visual answer to the question of how to represent many varying frequencies in one fixed coordinate system.

### 23.4 Four distinct expansion calendars

The lab keeps four related but non-equivalent birth rules separate:

1. **Global factor-depth birth**
   \[
   2^L.
   \]
   This is the first integer at which total factor depth \(L\) can occur.

2. **Prime-power lane birth**
   \[
   q^a.
   \]
   This is the first strike of atlas cell \(A(i,a)\).

3. **Independent base-clock necessity**
   \[
   q^2.
   \]
   Beginning here, the \(q\)-clock can remove a composite that need not already be removed by a smaller prime clock.

4. **LPF-branch layer birth**
   \[
   p\,2^{L-1}.
   \]
   This is the first possible integer in largest-prime-factor branch \(p\) on total factor layer \(L\). Its exact branch-layer envelope remains
   \[
   p2^{L-1}\le n\le p^L.
   \]

These events describe different ways the overall structure gains a new possibility or required dimension.

### 23.5 First laboratory implementation

Created file:

`prime-power-coordinate-atlas-gap-state-lab_v1.html`

Default build:

- endpoint \(N=200{,}000\);
- current integer \(n=72\);
- complete atlas contains 17,984 prime columns and 18,120 valid prime-power cells;
- all columns exist in the dictionary, but the display is paged so the map remains readable;
- the first version permits builds through 2,000,000.

The opening example is intentionally

\[
71\to72\to73,
\qquad
72=2^3\cdot3^2.
\]

At \(n=72\), the cells for \(2,2^2,2^3,3,3^2\) strike simultaneously. Stepping to 73 removes every proper base-clock strike and reaches the next prime boundary.

### 23.6 Linked pages and controls

#### Page 1 — Coordinate atlas

- fixed prime-index columns and exponent rows;
- clickable cells with coordinate, period, frequency, birth, remainder, phase, previous strike, and next strike;
- page-through controls for earlier and later prime columns;
- direct jump to any prime column;
- scopes for all primes, the exact coverage bank, the compressed \(q\le97\) bank, or only factors appearing in the current gap;
- selected-prime family view showing all powers of one prime as synchronized phase tracks;
- previous / next integer, previous / next prime, and automatic stepping;
- factor chips that jump from the current revealed factorization to its atlas coordinates.

#### Page 2 — Gap fingerprint

- one matrix column per integer in the current prime-to-prime gap;
- one row per chosen atlas lane;
- exact base-clock and higher-power strikes;
- click-through from any matrix cell to the integer and atlas coordinate;
- selectable factor-support profile across the gap;
- previous / next gap navigation;
- retained completed-gap skeleton information.

#### Page 3 — Growth and births

- logarithmic timeline for all four birth calendars;
- clickable event table centered around the current integer;
- current global depth state, born-lane count, and independently required base-clock count;
- selected branch's exact first and upper bounds across layers.

#### Page 4 — Coordinate key and exports

The lab exports:

- complete atlas CSV;
- current integer phase-fingerprint CSV;
- current gap matrix CSV;
- current revealed gap-skeleton CSV;
- complete current-state JSON;
- gap matrix PNG;
- gap skeleton PNG.

### 23.7 Revealed truth versus causal phase state

A top-level toggle controls whether the current factorization and gap skeleton are shown.

With truth enabled, this is an exploratory structural laboratory. The completed skeleton can be inspected for alternative organizations, compressed features, and visual regularities.

With truth disabled, the factorization and skeleton are hidden while the phase atlas remains available. This does not yet create a fully separated prediction process, but it makes the information boundary visible and testable during design.

The phase atlas is legal pre-reveal information because it follows from the current integer and previously defined periods. The completed target factorization is revealed information.

### 23.8 Construction-driven versus experimental content

Exact and construction-driven:

- atlas addresses \(A(i,a)\);
- every period and frequency;
- every phase and strike;
- every birth calendar;
- every revealed factorization;
- the alternating factor-occurrence skeleton coordinates.

Still experimental:

- whether the atlas feels natural enough to become a useful counting language;
- whether completed gap skeletons suggest a better coordinate organization;
- whether phase neighborhoods can be compressed without losing the next prime hole;
- whether gap-level fingerprints transfer across scale;
- whether a different layout reduces computation or makes a predictive relation visible.

The alternating factor-skeleton side placement creates part of its own visual symmetry. Any apparent symmetry in that projection must therefore be tested against the construction.

### 23.9 Validation status

The first build was exercised programmatically in a browser engine. Checks included:

- exact build through 200,000;
- correct prime and composite labels for 71, 72, and 73;
- correct gap identification \(71\to73\);
- exact strike of \(A(1,3)=2^3=8\) at 72;
- working integer navigation;
- working cell selection;
- working gap-matrix construction;
- working birth tables;
- no JavaScript page errors during the tested interactions.

### 23.10 Immediate next stage

Do not begin another prediction sweep before visually auditing this first atlas.

The next task is to step through representative sequences, click cells, watch phase motion, and decide whether:

- prime index should remain the horizontal address;
- exponent should remain the vertical address;
- phase should be shown as a track, a circular hand, or both;
- the gap skeleton should remain at the side, move underneath, or become a transparent background layer;
- a second projection by log period or frequency should be added;
- the complete gap fingerprint suggests a more compact gap coordinate.

The architecture is intentionally revisable. Testing the representation is part of discovering the process rather than merely illustrating a process chosen in advance.

---

### Recorder update ledger — Coordinate Atlas v1

- Added the permanent coordinate \(A(i,a)\leftrightarrow q_i^a\).
- Added the moving phase-state definition and visual track interpretation.
- Separated four different structural birth calendars.
- Recorded the new four-page standalone atlas laboratory.
- Preserved completed gap skeletons as optional exploratory information.
- Added a truth-visibility toggle to distinguish descriptive and causal views.
- Recorded default build size, opening example, export schemas, and validation checks.
- Next planned work: visual audit and coordinate redesign before renewed prediction testing.

---

## 24. Foundations-paper checkpoint: Prime-Power Recurrence Atlas v0.2

### 24.1 Conceptual refinement

The central object is now stated as a **Prime-Power Factor Matrix** inside a broader **Prime-Power Recurrence Atlas**.

For the ordered primes

\[
2=p_1<p_2<p_3<\cdots,
\]

coordinate address

\[
A(i,a)
\]

means the permanent cell belonging to prime base \(p_i\) and exponent level \(a\). The address is static. Its state at integer \(n\) is

\[
X_{i,a}(n)=\mathbf 1_{p_i^a\mid n}.
\]

The decisive distinction is that the cell does not refer only to the isolated pure power \(p_i^a\). It is active in **every integer containing that prime power**. Thus the \(2^2\) cell is active at \(4,8,12,16,20,\ldots\), regardless of the other factors present.

### 24.2 Exact recurrence and nestedness

Each coordinate has exact fundamental period \(p_i^a\):

\[
X_{i,a}(n+p_i^a)=X_{i,a}(n).
\]

Higher exponent cells are nested inside lower cells of the same prime column:

\[
X_{i,a+1}(n)\le X_{i,a}(n).
\]

These are exact divisibility facts, not numerical observations.

### 24.3 Integer fingerprints

The active-cell set

\[
\mathcal F(n)=\{(i,a):X_{i,a}(n)=1\}
\]

is the integer's prime-power fingerprint. The height of prime column \(p_i\) is exactly \(v_{p_i}(n)\), so the fingerprint uniquely reconstructs the prime factorization. Its total occupied-cell count is

\[
|\mathcal F(n)|=\Omega(n).
\]

Weighting each active coordinate by the logarithm of its base prime gives

\[
\log n=\sum_{(i,a)\in\mathcal F(n)}\log p_i
      =\sum_{d\mid n}\Lambda(d).
\]

### 24.4 Universal composite propagation

For any integer \(m\), its entire fingerprint is contained in the fingerprint of \(n\) exactly when \(m\mid n\):

\[
\mathcal F(m)\subseteq\mathcal F(n)
\quad\Longleftrightarrow\quad
m\mid n.
\]

Therefore the complete required factor pattern of \(m\) recurs inside every multiple of \(m\), with exact period \(m\). Additional coordinates may also be active at those multiples. This is the precise meaning of a recurring composite subpattern.

### 24.5 Prime boundaries

Before a new base coordinate is introduced at integer \(n\), inspect all previously established prime-base coordinates. An integer is prime exactly when no earlier prime base divides it. After recognition, the prime begins its own new base column. Thus:

- a composite reuses an intersection of earlier coordinate families;
- a prime is a new-base birth event in the expanding atlas.

This is an exact characterization and recursive generation rule. With the complete required base set over a bounded interval it is equivalent in work to ordinary divisibility elimination; the open problem is useful compression.

### 24.6 Fixed visual convention

The paper and future atlas revisions use a conventional lower-left origin:

- prime bases increase from left to right;
- exponent or power levels increase from bottom to top;
- each coordinate cell has one permanent address;
- cells retain a fixed comparison size;
- when the atlas grows, the view expands, scrolls, or pans rather than shrinking all cells to fit a fixed rectangle.

This layout is intended to make successive integers, fingerprints, and atlas growth directly comparable.

### 24.7 Gap skeleton role

A completed prime gap can be represented as an ordered array

\[
G_{i,a,t}(P,Q)=X_{i,a}(P+t),
\qquad 1\le t\le Q-P-1,
\]

for consecutive primes \(P<Q\). This retains the ordered sequence of prime-power fingerprints between the two prime boundaries.

Completed gap skeletons are post-reveal descriptions, but they remain legitimate exploratory information. Past completed skeletons may suggest alternative coordinates, reusable motifs, or compressed states. Their predictive value must be tested prospectively and against matched controls.

### 24.8 Preliminary paper v0.2

A 20-page discussion draft was prepared under the title:

**Prime-Power Recurrence Atlas: Integer factor fingerprints, composite propagation, prime boundaries, and preliminary spectral evidence.**

It contains:

- a precise coordinate definition;
- the lower-left, fixed-cell matrix convention;
- exact recurrence and nesting results;
- unique fingerprint reconstruction;
- universal composite propagation;
- prime-boundary and new-base characterization;
- branch ownership and the sharp branch-layer envelope;
- gap fingerprints and gap skeletons;
- the audited reduced-base experiment through 97;
- the finite zeta-zero convergence and frozen-threshold experiments;
- a strict separation of exact, empirical, and open claims;
- a proposed prospective research program and questions for mathematical review.

The manuscript intentionally avoids relying on the terms “clock” and “sieve.” It presents the system in recurrence-coordinate, factor-fingerprint, and prime-boundary language.

### Recorder update ledger — foundations paper v0.2

- Promoted recurrence through all containing composites to the central theorem-level distinction.
- Defined the Prime-Power Factor Matrix and Prime-Power Recurrence Atlas terminology.
- Fixed the visual coordinate convention at the lower-left with primes rightward and exponents upward.
- Required fixed-size cells and expanding/scrolling views rather than rescaling the atlas cells.
- Preserved completed gap skeletons as possible future compression information.
- Recorded the first revised foundations manuscript and its exact/empirical/open claim boundaries.


## Interface checkpoint — Prime-Power Factor Matrix Lab v2.1

This update changes presentation and navigation only; it does not alter the mathematical definitions or experimental claims.

- The boundary-test start field now accepts any integer and automatically resolves a composite entry to the greatest prime not exceeding it. The requested value and normalized prime are both preserved in the locked export.
- The fixed matrix's left exponent-label column and bottom prime-base row now have stronger outlines and a distinct axis background.
- Inactive matrix cells are lighter neutral gray-blue, making occupied prime-power coordinates more visible without changing cell dimensions.
- The Gap fingerprint page now places the coordinate matrix, selected-integer details, and a synchronized completed gap skeleton side by side on wide displays. The skeleton remains marked as revealed descriptive information.
- Automated browser validation passed for the adjusted starting-prime rule, added skeleton canvas, fixed orientation, and JavaScript execution through a build of 200,000.


## 25. Interface checkpoint — Gap Sequence Growth Lab v2.2

The Prime-Power Factor Matrix laboratory now includes a dedicated **Gap sequence growth** page. Its purpose is to expose the ordered development of one complete prime-to-prime interval before introducing a more complicated prediction model.

### 25.1 Sequence orientation and navigation

For consecutive primes \(P<Q\), the page displays

\[
P,\ P+1,\ldots,\ Q
\]

as a vertical sequence with offset 0 at the bottom and the right-hand prime at the top. Previous and next gap controls, single-integer stepping, start/end controls, and animation are provided. Previously reached rows remain visible during growth.

The selected step is

\[
t=n-P,\qquad 0\le t\le Q-P.
\]

### 25.2 Synchronized representations

Three views advance together:

1. **Ordered integer rows.** Each row records the integer, its offset, revealed factorization, \(\Omega(n)\), and \(\omega(n)\).
2. **Current integer structure.** The expanded occurrence tree shows one point per prime-factor occurrence, while the linked Prime-Power Factor Matrix shows nested exponent coordinates. One gap-wide scale is retained as the integer changes.
3. **Accumulated gap forest.** Every integer keeps a fixed-width column. Reached factor trees remain visible, the current tree is highlighted, and the unreached future may be ghosted or hidden.

This unfolded forest is distinct from the collapsed gap skeleton. The unfolded view preserves integer order; the collapsed view overlays occurrence slots into a common projection.

### 25.3 Row measurements

For each integer \(n\) in the inclusive gap, the page records:

- total factor occurrences \(\Omega(n)\);
- distinct prime bases \(\omega(n)\);
- higher-power activation count \(\Omega(n)-\omega(n)\);
- greatest exponent in any prime column;
- SPF and LPF;
- inherited base-coordinate count;
- von Mangoldt source weight \(\Lambda(n)\);
- logarithmic composite support
  \[
  \log\!\left(\frac{n}{\operatorname{LPF}(n)}\right),
  \]
  with value zero at primes;
- any prime-power coordinate whose first integer is \(n\).

The equality

\[
\Omega(n)-\omega(n)
\]

counts the occupied higher-power cells above the base row. It therefore separates first-level prime-base activations from repeated-factor depth.

### 25.4 Whole-gap totals

For the interior composite rows \(P<n<Q\), the page computes:

\[
M_{\Omega}(P,Q)=\sum_{P<n<Q}\Omega(n),
\]

\[
M_{\omega}(P,Q)=\sum_{P<n<Q}\omega(n),
\]

and

\[
M_{\mathrm{power}}(P,Q)
=\sum_{P<n<Q}\bigl(\Omega(n)-\omega(n)\bigr).
\]

It also records mean and maximum factor depth, greatest exponent, distinct prime bases used, logarithmic support mass, and the inclusive prime-power source weight

\[
M_{\Lambda}[P,Q]=\sum_{P\le n\le Q}\Lambda(n).
\]

A normalized cumulative chart compares the fractions of the completed-gap totals accumulated by each selected step. These quantities are currently descriptive candidate statistics. They are not yet claimed as Riemann-hypothesis invariants or predictive laws.

### 25.5 Structural distributions

The completed-gap panel summarizes:

- the number of interior rows on each \(\Omega\)-level;
- the most frequent prime-factor occurrences;
- prime-power coordinate births within the inclusive gap.

This supports the search for section signatures, recurring gap types, and possible lossless or predictive compressions.

### 25.6 Exports

The laboratory exports:

- one exact row-statistics CSV;
- one cumulative-timeline CSV;
- a complete gap-growth JSON package;
- the accumulated forest as PNG.

The general current-state JSON now embeds the gap-growth package as well.

### 25.7 Cell-label refinement

Long dormant-coordinate labels no longer appear as one clipped line. They are displayed as two stacked lines, for example:

```text
introduced at
131,072
```

The active and new-base messages use the same stacked arrangement, leaving more room for the phase marker and large coordinate values.

### 25.8 Validation

Static DOM and JavaScript checks found unique element identifiers and no unresolved element references. Independent arithmetic checks verified:

- for gap \(71\to73\), the single interior integer \(72=2^3\cdot3^2\) has \(\Omega=5\), \(\omega=2\), and \(\Omega-\omega=3\);
- for gap \(23\to29\), the five interior integers have totals
  \[
  \sum\Omega=14,\qquad \sum\omega=8,\qquad
  \sum(\Omega-\omega)=6.
  \]

Automated render-path checks generated the sequence, summary cards, row table, current structure, accumulated forest, and export objects without an exception.

### Recorder update ledger — Gap Sequence Growth v2.2

- Added a prime-to-prime animated growth view with the left prime at the bottom and the next prime at the top.
- Preserved every reached integer structure during animation.
- Added an unfolded, scrollable factor-occurrence forest with fixed integer-column width.
- Added exact per-integer and whole-gap measurements.
- Added normalized cumulative structure curves and completed-gap distributions.
- Added dedicated row, cumulative, JSON, and image exports.
- Retained an explicit warning that completed-gap measurements are descriptive until a compression rule is frozen and tested prospectively.


## 26. Paired factor--exponent tree and continuous gap traversal (Lab v2.4)

### 26.1 Why a second tree encoding was added

The expanded occurrence skeleton remains mathematically useful because it displays one point for every prime-factor occurrence. However, it does not directly use one side of the tree for a prime base and the opposite side for its exponent. A second, losslessly compressed projection was therefore added to the Gap sequence growth tab.

For an integer

\[
n=\prod_{j=1}^{\omega(n)} q_j^{a_j},
\]

with distinct prime factors ordered increasingly, the paired tree assigns the term \(q_j^{a_j}\) the two horizontal addresses

\[
x_{\mathrm{left}}=-a_j,
\qquad
x_{\mathrm{right}}=+j.
\]

The right node is labeled by the actual prime base \(q_j\). The left node records exponent \(a_j\). Both connect through a center spine labeled by the integer's largest-prime-factor seed.

Examples:

- \(3^4\) is represented by a left node at \(-4\) and a right node at \(+1\) labeled 3.
- \(80=2^4\cdot5\) has the paired terms \(2^4@(-4,+1)\) and \(5^1@(-1,+2)\).
- \(36=2^2\cdot3^2\) has two left nodes at the same exponent coordinate \(-2\). They are displaced vertically by a small categorical amount so both remain visible without changing their horizontal coordinate.

The paired tree's vertical direction is therefore not an additional arithmetic magnitude. It separates distinct factor terms that would otherwise occupy the same horizontal address. The exact arithmetic content remains the ordered set \((q_j,a_j)\).

### 26.2 Relation to the expanded occurrence tree

The new paired tree is the default, but the previous occurrence view remains selectable.

- **Expanded occurrence tree:** one point per factor occurrence; its point count equals \(\Omega(n)\).
- **Paired factor--exponent tree:** one branch pair per distinct prime factor; its branch count equals \(\omega(n)\), while the left coordinate preserves each exponent exactly.

The two projections are losslessly interconvertible from the factorization and can be compared as possible compression geometries.

### 26.3 Continuous traversal through consecutive gaps

The Gap sequence growth animation now has an optional continuous mode. When enabled, the current integer advances by one throughout a gap. On reaching its upper prime, that same prime becomes the lower boundary of the following gap and the animation continues with the next integer. It stops only when paused or when the next complete gap lies outside the built range.

The scroll-follow mechanism remains confined to the two internal sequence panes; the document viewport does not move during playback.

### 26.4 Completed-gap navigation

Selecting Previous gap or Next gap now opens the requested gap at its upper prime rather than at its lower prime. Consequently every row in that selected gap is already in the reached state and its whole tree/forest is immediately displayed at full brightness. The user can still press Go to lower prime to replay the construction from offset zero.

### 26.5 Export additions

Per-integer sequence exports now retain a compact paired-tree field such as

```text
2^4@(-4,+1);5^1@(-1,+2)
```

and the center largest-prime-factor seed. The JSON package also records which tree encoding was selected and whether continuous traversal was enabled.

### 26.6 Validation

Automated browser checks verified:

- \(81=3^4\) maps to \((-4,+1)\) with right node 3;
- \(80=2^4\cdot5\) maps to \((-4,+1)\) and \((-1,+2)\);
- equal exponents in \(36=2^2\cdot3^2\) share horizontal coordinate \(-2\) while remaining separately visible;
- Next gap from the completed gap \(79\to83\) opens \(83\to89\) at 89;
- continuous playback crosses prime boundaries without stopping;
- no JavaScript page errors occurred in the tested paths.

### Recorder update ledger -- Factor Tree Lab v2.4

- Added the paired factor--exponent tree as the default Gap sequence growth encoding.
- Preserved the expanded factor-occurrence tree as an alternate view.
- Added coordinate-collision separation for equal exponents.
- Added continuous integer playback across successive prime gaps.
- Changed gap navigation to reveal the selected gap at its completed endpoint.
- Added paired-tree coordinates and settings to the research exports.

## 27. Prime-gap structural balance laboratory (Lab v2.5)

### 27.1 Motivation

The Gap sequence growth page made a new exact quantity visually accessible: the total active prime-power-coordinate mass carried by all integers in a completed prime gap. For a consecutive-prime interval

\[
P<Q,
\]

the interior \(\Omega\)-mass is

\[
M_\Omega(P,Q)=\sum_{P<n<Q}\Omega(n).
\]

Because \(\Omega(n)\) is exactly the number of active nested prime-power coordinates in the fingerprint of \(n\), the same total can be counted by integers or by coordinates:

\[
M_\Omega(P,Q)
=
\sum_p\sum_{a\ge1}
\#\{n:P<n<Q,\ p^a\mid n\}.
\]

This identity turns the completed gap forest into a measurable structural object rather than only a visualization.

### 27.2 Separate notions of balance

Lab v2.5 adds a dedicated **Gap balance** tab and keeps two questions distinct.

1. **Internal shape balance.** How is the completed observed mass ordered between the lower and upper prime?
2. **Absolute recurrence drift.** How does the observed cumulative amount compare with a smooth total implied by the permanent recurrence frequencies?

The first is normalized to begin and end at zero. The second generally is not.

### 27.3 Normalized cumulative mass and residual bridge

Choose a completed gap, a nonnegative row mass \(w_j\), and a row scope. The default scope uses only the interior integers. If there are \(m\) included rows, define

\[
x_j=\frac{j}{m},
\qquad
W=\sum_{j=1}^{m}w_j,
\qquad
F_j=\frac{\sum_{k=1}^{j}w_k}{W}.
\]

The normalized balance bridge is

\[
B_j=F_j-x_j.
\]

It satisfies

\[
B_0=0,
\qquad
B_m=0.
\]

Positive bridge values mean that the selected mass has accumulated earlier than a uniform row distribution. Negative values mean that it has accumulated later.

The tab reports:

- half-mass crossing \(x_{50}\);
- mass center of gravity;
- left and right mass shares;
- right-minus-left imbalance;
- maximum positive and negative bridge excursions;
- maximum absolute bridge;
- bridge RMS;
- signed bridge area.

Under the equal-width-bin convention used by the page,

\[
\int_0^1 \bigl(F(x)-x\bigr)\,dx
=
\frac12-c,
\]

where \(c\) is the mass center.

### 27.4 Available structural masses

The first release allows the same analysis to be repeated for:

- active prime-power cells \(\Omega(n)\);
- composite support \(\Omega(n)-1\);
- distinct prime bases \(\omega(n)\);
- higher-power cells \(\Omega(n)-\omega(n)\);
- logarithmic support \(\log(n/\operatorname{LPF}(n))\);
- the complete weighted fingerprint \(\log n\);
- the prime-power source weight \(\Lambda(n)\).

The page handles zero-mass cases explicitly. For example, an interior gap containing no pure prime power has zero \(\Lambda\)-mass, so a normalized \(\Lambda\)-bridge is undefined unless the row scope or gap is changed.

### 27.5 Exact recurrence-frequency expectation

For additive recurrence metrics, the page also computes a smooth rate from the permanent coordinate frequencies. Examples are

\[
\mu_\Omega(B)=\sum_{p^a\le B}\frac1{p^a},
\]

\[
\mu_\omega(B)=\sum_{p\le B}\frac1p,
\]

\[
\mu_{\mathrm{higher}}(B)=
\sum_{\substack{p^a\le B\\a\ge2}}\frac1{p^a},
\]

and

\[
\mu_\Lambda(B)=
\sum_{p^a\le B}\frac{\log p}{p^a}.
\]

For an interval of \(m\) consecutive included integers, the smooth expected amount is \(m\mu(B)\). The expectation-drift chart displays

\[
\text{observed cumulative mass}
-
\text{expected cumulative mass}.
\]

Unlike the normalized bridge, this curve need not return to zero. Its endpoint records the total finite-interval excess or deficit.

### 27.6 Atomic coordinate discrepancy

For one coordinate \(p^a\) in a consecutive interval \([A,B]\), the exact activation count is

\[
\left\lfloor\frac{B}{p^a}\right\rfloor
-
\left\lfloor\frac{A-1}{p^a}\right\rfloor.
\]

Its smooth count is

\[
\frac{B-A+1}{p^a}.
\]

The difference has magnitude less than one. Lab v2.5 lists this phase discrepancy for every born coordinate. Summing the actual coordinate counts reproduces the exact \(\Omega\)-mass, while summing the smooth counts gives the finite-coordinate expectation.

This supplies an exact decomposition of local mass excess into bounded phase discrepancies from the permanent prime-power coordinates.

### 27.7 Controls

The balance page includes:

- unrestricted order shuffles;
- parity-preserving order shuffles;
- 64, 128, or 256 deterministic repetitions;
- a 5th-to-95th percentile bridge envelope;
- the observed bridge's descriptive percentile relative to the controls.

These envelopes are controls for ordering effects, not formal confidence intervals. Stronger future controls should preserve additional small-prime residue structure.

### 27.8 Neighbor-gap scan and component decompositions

The page can scan 21, 51, or 101 nearby completed gaps using the same metric and row scope. It can plot:

- center-of-mass offset;
- half-mass crossing offset;
- maximum absolute bridge;
- total mass;
- recurrence-expectation excess.

The current gap is highlighted and each neighboring point is clickable.

Two linked tables decompose the selected mass by:

- prime base;
- exponent level.

Each table records the component's total mass, left and right shares, and its own center. A separate table lists the atomic \(p^a\) recurrence discrepancies.

### 27.9 One-half claim boundary

The normalized guide lines

\[
x=\frac12,
\qquad
F=\frac12
\]

belong to the rescaling of a finite prime gap. They are not identified with the real coordinate \(1/2\) in the Riemann hypothesis.

The potentially substantive research questions are instead:

- how close the half-mass crossing remains to the gap midpoint;
- how the bridge excursions scale with gap length and number size;
- whether real ordering remains unusual under controls that preserve modular structure;
- whether coordinate-level excesses exhibit a stable cancellation law;
- whether a compressed balance signature can help classify or predict future gaps.

### 27.10 Interface and exports

The v2.5 tab includes five large charts:

1. normalized cumulative mass against the diagonal;
2. the residual bridge;
3. the unaccumulated per-integer mass profile;
4. recurrence-expectation drift;
5. the chosen balance statistic across neighboring gaps.

It exports:

- per-integer balance CSV;
- complete balance JSON;
- neighboring-gap CSV;
- cumulative chart PNG;
- bridge chart PNG.

The general complete-state JSON now embeds the full current gap-balance package.

### 27.11 Validation

The built-in browser audit through \(N=200{,}000\) verified:

- gap \(271\to277\) has interior \(\Omega\)-mass 17;
- the normalized bridge begins and ends at zero;
- the sum of exact atomic coordinate activations also equals 17;
- the balance CSV contains one row for every included interior integer;
- the balance JSON records the same total and summary;
- all dedicated canvases render.

A second automated browser sweep exercised all 42 combinations of:

- seven mass definitions;
- two row scopes;
- three control modes.

No JavaScript errors were observed.

For the illustrative gap \(271\to277\), using interior \(\Omega\)-mass:

- total mass: 17;
- half-mass crossing: 0.4500;
- mass center: approximately 0.4765;
- left/right shares: approximately 52.9% / 47.1%;
- maximum absolute bridge: approximately 0.0941.

These values are an interface validation example, not a claimed universal law.

### Recorder update ledger -- Gap Balance Lab v2.5

- Added a dedicated prime-gap structural-balance tab.
- Separated normalized internal balance from absolute recurrence-frequency drift.
- Added half-mass, center-of-mass, left/right, bridge, and excursion measurements.
- Added parity-preserving and unrestricted order-control envelopes.
- Added exact coordinate-by-coordinate recurrence discrepancy accounting.
- Added prime-base and exponent-level balance decompositions.
- Added a clickable neighboring-gap scan.
- Added CSV, JSON, and PNG exports.
- Preserved the explicit warning that the normalized one-half is not automatically the Riemann-hypothesis one-half.


---

## 28. Factor–Power correspondence and gap half-balance checkpoint (2026-09-10)

### 28.1 Exact factor–power correspondence

For

\[
n=\prod_p p^{e_p},
\]

there is a canonical one-to-one map

\[
\text{copy }k\text{ of factor }p
\longleftrightarrow
p^k,
\qquad 1\le k\le e_p.
\]

Therefore, for every integer \(n\ge2\),

\[
\boxed{
\#\{\text{prime-factor occurrences in }n\}
=
\#\{\text{active nested prime-power coordinates at }n\}
=
\Omega(n).
}
\]

After compressing each prime column to its greatest active exponent,

\[
\boxed{
\#\{\text{distinct prime bases}\}
=
\#\{\text{exponent entries}\}
=
\omega(n).
}
\]

These identities hold row by row and hence after summing any finite set, prime gap, layer regime, or interval.

The integer 1 is the empty multiplicative origin:

\[
\Omega(1)=\omega(1)=0.
\]

It has no factor occurrences, no active prime-power coordinates, and no LPF.

### 28.2 Two different meanings of balance

The exact correspondence above is a **vertical balance inside each integer**. It does not imply a **horizontal balance across the positions of a prime gap**.

For consecutive odd primes \(P<Q\), the number of interior integers is odd. Write the interior as an earlier half, one center integer, and a later half. For a selected nonnegative mass \(w(n)\), define

\[
L=\sum_{\text{earlier half}}w(n),
\qquad
C=w(\text{center}),
\qquad
R=\sum_{\text{later half}}w(n),
\]

and

\[
\Delta=R-L.
\]

Nothing in unique factorization forces \(\Delta=0\) or \(\Delta>0\). Dividing the center mass equally between the two halves adds \(C/2\) to each side and leaves \(R-L\) unchanged.

The total cumulative structure grows because every new integer contributes nonnegative mass and the atlas gains new prime and power coordinates. Local later-half dominance is not required for that global growth.

For a monotone magnitude weight such as \(\log n\), every mirrored later integer is automatically larger than its earlier partner. A later-heavy result under that metric is construction-driven. The combinatorial masses \(\Omega\), \(\omega\), and \(\Omega-\omega\) are not monotone.

### 28.3 Preliminary half-gap scan through ten million

An independent linear-sieve scan tested all 664,577 completed odd-prime gaps through \(10,000,000\).

| Mass | Later-heavy | Earlier-heavy | Equal | Mean \(R-L\) |
|---|---:|---:|---:|---:|
| \(\Omega\) | 268,628 | 269,838 | 126,111 | -0.009470686 |
| \(\omega\) | 247,006 | 247,732 | 169,839 | -0.008108917 |
| \(\Omega-\omega\) | 263,413 | 264,001 | 137,163 | -0.001361768 |
| \(\log n\) | 605,597 | 0 | 58,980 twin-gap equalities | +0.000065431 |

Thus the strong statement “the later half is always heavier” is false for the factor-count metrics. The near equality of the two directional counts suggests a more defensible aggregate-balance question.

### 28.4 Exact coordinate decomposition of half imbalance

For \(\Omega\)-mass,

\[
\Omega(n)=\sum_{p^a\mid n}1.
\]

Hence the half difference decomposes exactly as

\[
\boxed{
R-L
=
\sum_{p^a}
\left(
\#\{\text{later appearances of }p^a\}
-
\#\{\text{earlier appearances of }p^a\}
\right).
}
\]

This makes it possible to separate short-period from long-period recurrence coordinates. A large period means a small frequency. Long-period coordinates are sparse in one short gap and can create one-sided tilts, while short-period coordinates recur more often and may cancel more locally.

### 28.5 New standalone laboratory

Created:

**PrimeMind Factor–Power Certificate & Gap Half-Balance Lab v1**

The lab contains:

1. **Integer certificate**
   - canonical occurrence-to-coordinate pairing;
   - expanded \(\Omega\) balance;
   - compressed \(\omega\) balance;
   - product and logarithmic reconstruction;
   - tamper demonstrations.

2. **Gap certificate**
   - row-by-row exact checks;
   - inclusive visual totals;
   - non-overlapping half-open \([P,Q)\) totals.

3. **Half balance & growth**
   - explicit earlier / center / later decomposition;
   - split-center shares;
   - raw and normalized \(R-L\);
   - mass center and cumulative bridge;
   - mirrored row-pair table;
   - several mass definitions.

4. **Coordinate decomposition**
   - one row per active \(p^a\) coordinate;
   - left, center, and right counts;
   - exact contribution to \(R-L\);
   - short/long period split.

5. **Bulk scan**
   - all completed odd-prime gaps through a selected endpoint;
   - later/earlier/equal counts;
   - running mean and distribution;
   - layer-regime summaries;
   - counterexample navigation;
   - compact CSV and JSON exports.

6. **Definitions & exports**
   - exact versus empirical claim boundary;
   - current integer, gap, half-balance, coordinate, and bulk exports.

### 28.6 Validation

The final browser test through \(10,000,000\) produced:

- 664,579 primes;
- 664,577 completed odd-prime gaps;
- exact agreement with the independent C++ half-balance scan;
- \(72=2^3\cdot3^2\) with five factor occurrences and five active nested coordinates;
- \(1\) as the empty \(0\leftrightarrow0\) state;
- immediate certificate failure after deliberately deleting one factor occurrence;
- no JavaScript errors in the tested views.

### 28.7 Current interpretation

**Exact:** factor occurrences and active nested power coordinates are canonically paired; distinct bases and exponent entries are canonically paired; these equalities add over finite sections.

**Not exact:** the center of every prime gap, or the two strict halves of every prime gap, need not carry equal structural mass.

**Open:** whether the aggregate directional imbalance tends to zero, obeys a useful scale bound, or admits a predictive signature after decomposition by period, base, exponent, and layer regime.

### Recorder update ledger — Factor–Power Certificate & Half-Balance Lab v1

- Formalized the expanded and compressed one-to-one correspondences.
- Added the integer 1 as the empty origin.
- Separated vertical factor–power balance from horizontal gap balance.
- Added strict earlier/center/later and split-center conventions.
- Falsified the “later half always larger” claim for \(\Omega\), \(\omega\), and higher-power mass through ten million.
- Added exact coordinate-by-coordinate decomposition of the half difference.
- Created a standalone certificate, tamper, half-balance, coordinate-tail, and bulk-scan laboratory.