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+# P4 — A matching-odd finite-size signal, and the failure of scalar closure
+
+Draft manuscript notes for **Paper 1** of [issue #644](https://github.com/LightChainr/Matching-One/issues/644).
+Portfolio track **P4**, following the `p2`/`p3` manuscript convention.
+
+| File | Role |
+|---|---|
+| `README.md` (this file) | the draft: claims, evidence, limitations, omissions |
+
+**Status of this document.** A first-pass writing draft only. It promotes nothing,
+re-scores nothing, and computes nothing; `docs/STATUS.md` was not touched and the claim
+levels quoted below are its current `main` content, cited as-is. Issue #644 stays open.
+No venue is selected. Unlike `p2`/`p3` there is **no generated `tables.md` and no
+evidence-assembly script yet** — every number below is carried with the artifact path and
+JSON field where its full-precision value is committed, which is the substitute contract
+until a generation pipeline is written (see §10). All paths are relative to the repository
+root.
+
+## 1. Abstract (draft)
+
+We report a real matching-odd orientation signal in finite 2D percolation lattices,
+established by a two-block independent synthesis, and we report that every attempt to
+close the observed finite-size structure with a *single scalar* — one multiplier, one
+width, or one pure exponent — fails. Two preregistered, independent 500M-replica evidence
+blocks jointly reject a global zero-effect description
+(`chi2 = 31.1857355515/4`, `p = 2.81e-6`), while frozen fixed-H4 predictions remain
+compatible (`3.4622795373/4`, `p = .484`). On a frozen norm-5 harmonic discrimination,
+the spin-4 alias H4 beats its adversarial alias H12 and its control H8 decisively.
+A frozen two-parameter finite-size center-slope correction predicts a held-out doubling
+lineage to within `0.67` standard errors where the bare asymptotic multiplier misses by
+more than 22; but the same held-out curve rejects one-multiplier transfer of the
+three-level thermal-even profile (`9.3520/2`, `p = .0093`), a scalar width-collapse
+over the full covariance fails (`24.5004/10`), and the pure exponent law
+`P4[S'] ~ N^(-5/4)` is prospectively falsified on new geometry (`52.71634/2`).
+The contribution is the combination: the signal is real, a frozen finite-size transfer
+works, and *scalar closure in any of its three forms does not*. We state the correlated
+block accounting explicitly, and we make no claim about `p_c`.
+
+## 2. Scope and non-goals
+
+- **No threshold claim.** Nothing here addresses `p_c`, its value, or its identification.
+  The ticket is explicit that this paper "does not need to explain `p_c` and should not
+  try."
+- **No unified-theory claim.** The paper does not assert the continuum operator, the state
+  dimension, or an LCFT module identification. The live mechanism state is low-rank /
+  multicomponent and that is where the paper stops.
+- **No new computation.** All numbers are already scored and committed; this draft adds
+  arrangement and wording only.
+- What the negatives are: prospective falsifications of frozen scalar-closure hypotheses,
+  made countable by the frozen-prediction discipline of §5. That discipline is the
+  paper's methodological spine, not an appendix.
+
+## 3. Observables and the exact structure that defines them
+
+**The observable.** From the threshold-rank engine's long-form batch histograms, the
+intrinsic-center p0 where the direction-average matching function vanishes, and the
+primal/matching pair `S = (R_G + R_hat)/2`, `D = (R_G − R_hat)/2` (with
+primes = `p`-derivatives at the center, from the exact beta-density identities
+`dR_hat(1−p)/dp = −E[beta_pdf(K_minus)]`), the spin-4 orientation contrast is
+`P4[X] = (X(θ1) − X(θ2))/Δcos(4θ)` — the first-minus-second orientation pairing is the
+matching-odd channel. Definitions and frozen parity powers (`P4[S] ~ N^-1`,
+`P4[D] ~ N^-13/8`, `P4[D'] ~ N^-5/8`, `P4[S'] ~ N^-5/4`) are committed in the analyzer
+header, `scripts/analyze_matching_parity_derivatives.py`. The primary target statistic of
+§5.1 is the matching-odd orientation contrast of the wrapping-channel matching function
+(threshold-rank engine, `first − second` in frozen genealogy order).
+
+**Exact channel correction (spine, not a footnote).** For the complementary
+primal/matching construction on the torus, cross and either channels obey the exact
+topological exchange `S_either = 1 − S_cross`, hence for orientation contrasts
+
+```text
+DeltaS_cross = -DeltaS_either        (exact, no fit)
+```
+
+recorded with its corrected no-refit score (`0.5700315436/2`, residual z `+0.6672,
+-0.1189`) in `results/server-20260828/P43-heldout-fullcurve-500m/analysis/channel_map_corrected_DeltaS.json`
+and `notes/issue43-cross-either-channel-map-erratum.md`. The odd-channel companion map has
+scale `+1` (`D_either = D_cross`), validated as an executable gate before scoring —
+`notes/norm5-harmonic-primary-scorer.md`, gate artifact
+`predictions/norm5_harmonic_semantic_gate_20260829.yaml`. Without these exact maps the
+apparent sign behaviour of the matching-even sector is misread (the erratum's consequence:
+an interpretation fix, not a data problem — #108's two-sector reading reclassified to
+prospective *compatibility* after the map).
+
+**Finite Russo identity (spine).** On a finite periodic quotient, the matching slope is
+exactly a sum of two nonnegative total pivotal masses:
+
+```text
+M'(p) = sum_v P_p(Piv_G(v)) + sum_v P_{1-p}(Piv_hat(v))
+```
+
+with `M = R_G(p) − R_hat(1−p)`; this is a chain-rule fact, not a scaling assumption.
+Derivation and three independent exact verification paths:
+`notes/pivotal-russo-finite-volume.md`, `scripts/exact_pivotal_russo.py`; the committed
+bridge analyses use it as a definition —
+`results/local-20260829/P100-pivotal-h4-bridge/score.json` (`definitions.Mbar_prime`:
+"exact total pivotal mass by Russo") — and the companion marked-pivotal analysis enforces
+the control role: `russo_control_role: "mu0 is a regression control, not independent
+evidence"` (`results/local-20260829/P100-marked-pivotal-h4/analysis/score.json`).
+
+**Frozen predictions.** Every falsification in §6 was scored against a prediction frozen
+before its target was revealed:
+
+```text
+predictions/gaussian_norm5_harmonic_discrimination_20260828.yaml   frozen_at 2026-08-28,
+    status preregistered_before_child_runs (H4/H12/H8 multipliers exact rational)
+predictions/p48_sprime_correction_20260828.yaml                    status frozen_before_target_reveal
+predictions/p49_slope_two_sector_145_290_20260828.yaml             status frozen_before_fullcurve_N290_reveal
+```
+
+This is what makes a negative result a *prospective falsification* rather than a bad fit
+(GOVERNANCE §2C).
+
+## 4. The evidence blocks, and counting one block once
+
+GOVERNANCE §2E: roots, slopes, derivatives, quantiles and score modes taken from the same
+histograms are all useful, and all are the *same block*; adding them as independent
+evidence inflates the conclusion. The paper's block inventory:
+
+| block | content | primary score |
+|---|---|---|
+| **I** | `issue43_n185_n265_500m_histograms` — held-out N=185/265 full curves, disjoint counter domains | matching-odd `DeltaM` fixed-H4 / zero-effect |
+| **II** | `issue57_norm5_production` — N=325/425 norm-5 harmonic discrimination, 500M/orientation | same, in its own channel |
+| **III** | `P50` N=145 → 290 full curve, independent seeds | three-level transfer, slope, channels |
+
+The joint two-block claim (Block I + II) is the Issue #212 synthesis, whose artifact
+freezes the independence contract (`raw_data_groups` distinct,
+`joint_covariance: block_diagonal`, scope "no other ledger views enter"):
+`results/evidence-ledger/issue212-matching-odd-synthesis.json`.
+
+Every other score in this paper is a *view of one block*, and the draft says so at each
+use: the H4/H12/H8 discrimination and the width-collapse score both live in Block II (the
+thermal-jet report states its secondary scores "reuse the primary raw curves and are not
+an additional independent evidence block"); the P48 `S'` pure-law score reads the intrinsic
+center analyses of Block I's own runs (`results/server-20260828/P48-new-geometry-score/REPORT.md`
+names its targets `analysis/n185.p48.json`, `analysis/n265.p48.json` from
+`P43-heldout-fullcurve-500m`); the central-contrast and channel tables of Block III are
+diagnostics, not evidence rows. There is no number in this paper that is the product of
+four confirmations; the honest headline count is two independent primary blocks plus one
+independent third block carrying falsifications.
+
+## 5. Positive results
+
+**5.1 The signal exists (two-block, primary).**
+
+```text
+zero effect:   chi2 = 31.18573555150965 / 4,  p = 2.805595267905808e-6
+fixed H4:      chi2 = 3.4622795373044295  / 4,  p = 0.48363695393249573
+```
+
+Field paths: `joint_scores.zero_effect.chi_square`,
+`joint_scores.zero_effect.chi_square_survival_p`, `joint_scores.fixed_H4.*` in
+`results/evidence-ledger/issue212-matching-odd-synthesis.json`; prose
+`notes/issue212-matching-odd-synthesis.md`. Direct predictive comparison: NLPD
+`fixed_H4 − zero_effect = -13.806418789729096` nats (field
+`predictive_comparison.delta_nlpd_fixed_H4_minus_zero_effect`); the artifact records
+`preferred_lower_nlpd: fixed_H4`. The chi-square *difference* is deliberately not treated
+as a likelihood-ratio statistic (frozen covariances need not agree).
+
+Ledger level cited from `main` without promotion: `docs/STATUS.md` line 25, "Square-site
+matching-odd orientation signal exists — C3".
+
+**5.2 Frozen H4 transfers; tested aliases are disfavored (Block II).**
+Fixed, parameter-free multipliers recomputed from exact rational angular factors and
+`5^(-13/8)`:
+
+| model | harmonic | chi2 / df | survival |
+|---|---|---:|---:|
+| H4 (primary) | `cos_4_theta`, target `-14/25 · 5^(-13/8)` | 0.4163 / 2 | 0.8121 |
+| H12 (adversarial alias) | `cos_12_theta`, `23506/15625 · 5^(-13/8)`, opposite child sign | 35.1931 / 2 | 2.280e-8 |
+| H8 (even-m control) | `cos_8_theta`, `-1054/625 · 5^(-13/8)` | 16.0120 / 2 | 3.335e-4 |
+| zero effect | — | 1.7764 / 2 | 0.4114 |
+
+Source: `results/server-20260829/P57-norm5-500m/primary_score.json` (models table,
+full-precision `chi_square` fields `0.41630376401835489631`, `35.193078878051031476`,
+`16.012022412848537938`), prose `results/server-20260829/P57-norm5-500m/REPORT.md`,
+frozen contract `predictions/gaussian_norm5_harmonic_discrimination_20260828.yaml`.
+H4 residuals `+0.645σ, −0.021σ`; H12 residuals `−4.612σ, −3.770σ`.
+`docs/STATUS.md` line 27: C3.
+
+**The honesty clause this section must keep:** the two target points alone do **not**
+reject a zero child effect (row 4 survives at `0.4114`). §5.2 supports the frozen H4
+*alias choice*; it is not an independent discovery of a nonzero H4 amplitude — the
+discovery claim is §5.1's, on two blocks. The REPORT says this in its own voice and the
+paper repeats it.
+
+**5.3 A frozen finite-size correction predicts a held-out lineage (Block III).**
+The two-parameter center-slope correction (scalar + frozen H4), trained only on
+`65 → 130` and `85 → 170`, predicts the held-out `145 → 290` slope:
+
+```text
+frozen scalar+H4 corrected slope:  signed z = -0.6660797543793089  (p = 0.50536)
+raw 2^(3/8) asymptotic baseline:   signed z = -22.690348726453408  (p = 5.58e-114)
+```
+
+Fields `slope.frozen_scalar_plus_H4_correction.signed_z` and
+`slope.raw_asymptotic_baseline.signed_z` of
+`results/server-20260829/P50-n145-n290-fullcurve/analysis/score.json`; table
+`results/server-20260829/P50-n145-n290-fullcurve/REPORT.md`; frozen prediction
+`predictions/p49_slope_two_sector_145_290_20260828.yaml`.
+`docs/STATUS.md` line 30: C3. This is the paper's cleanest positive finite-size transfer:
+a prediction with **zero refitted parameters** on the target, missing by 22 standard
+errors without the correction and by less than one with it.
+
+## 6. The failure of scalar closure (three forms, prospectively)
+
+**6.1 One multiplier does not close the three-level curve (Block III).**
+Joint `DeltaM` at `u = 0, 0.025, 0.05` under the frozen multiplier:
+`chi2 = 9.352003684819664 / 2`, `p = 0.009316` (field
+`primary_deltaM_transfer.chi_square` / `.chi_square_survival`,
+`results/server-20260829/P50-n145-n290-fullcurve/analysis/score.json`).
+`docs/STATUS.md` line 29: C3 negative.
+
+Where the failure lives, and why it is a shape failure and not an amplitude failure:
+the central contrast residual is only `−1.385` standard errors on its own
+(observed ratio `−0.52864747` vs frozen `−0.32420989`); rejection comes from the
+strongly-correlated three levels resolving a small deformation. Residual-correlation
+eigenvalues `1.23785e-13, 1.76308e-6, 2.99999824` give numerical rank 2 at the frozen
+cutoff `1e-10`; the amplitude-like mode contributes `≈ 1.9243` of the chi-square, the
+active shape mode `≈ 7.4277` (REPORT.md, "Where the joint DeltaM failure lives").
+The result does not overturn §5; it shows a single multiplier does not transport the
+local thermal-even curve *shape*.
+
+**6.2 One width does not collapse the thermal jet (Block II).**
+Full-covariance norm-5 width-collapse score:
+`chi2 = 24.500424925937486902 / 10`, survival `0.006377` (field
+`primary_width_collapse.score.chi_square`,
+`results/server-20260829/P57-norm5-500m/thermal_jet_score.json`; table
+`results/server-20260829/P57-norm5-500m/THERMAL_JET_SCORE.md`). Ordered follow-ons on
+the same block: width-corrected `q=2` analytic (`c=8/5`) `22.2386/10`, `p = .013934`;
+width-corrected rank-2 Jordan (`c = log5/log2`) `17.0513/10`, `p = .073237`.
+`docs/STATUS.md` line 32: C2 negative (scalar width); line 33 records that Jordan is
+*compatible, not identified*.
+
+**6.3 One pure exponent does not close `S'` (Block I).**
+The parity rule with `x = 21/4` and `y_t = 3/4` gives `P4[S'] ~ N^(-5/4)`; the
+scaled `Y_N = N^(5/4) P4[S']` frozen pure law (`A0 = 1.9434247576878727 ±
+0.0766048795577632`, trained on N=65/85/130) scores on the new N=185/265 geometries
+`chi2 = 52.71633588357711 / 2` (field `channels.P4_S_prime.chi_square`,
+`results/server-20260828/P48-new-geometry-score/score.json`; zero-model
+`1278.555`; marginals `+5.04σ` and `+6.61σ`).
+`docs/STATUS.md` line 31: C3 negative, "prospectively falsified".
+The three sibling channels of the same frozen score survive their pure laws
+(`P4[S]` `1.139/2`, `P4[D]` `0.281/2`, `P4[D']` `0.088/2`) — `S'` is the unique
+clear pure-law failure, so the falsification is specific to the matching-odd
+derivative channel, not to finite-size laws in general.
+The exponent *arithmetic* is not the broken part (the parity-rule derivation is in
+`notes/p48-sprime-correction-discriminator.md`); the constant-amplitude *closure* is.
+
+## 7. Where the mechanism boundary actually sits
+
+The three scalar failures are not generic. Within Block III, the projected derivative
+channels split sharply: `P4[D']` transfers under its frozen multiplier to `−0.009σ`
+(`p = 0.99275`) while `P4[S']` misses at `+2.70σ` (`p = 0.00703`) — REPORT.md,
+"Correlated P4 diagnostics". One projected direction of the derivative response is
+already a near-exact eigenchannel; its companion is not governed by the same scalar law.
+That is the evidence for treating the response as a small non-diagonal (Jordan-type)
+block, and it is the same ordering the independent norm-5 diagnostics found (Jordan
+`17.05/10` vs `q=2` `22.24/10`, improvement `5.19`, explicitly a ranking diagnostic
+between non-nested fixed models, not a likelihood-ratio test).
+
+The frozen correction models for the failed `S'` channel were themselves preregistered
+for N290 and both survive there (`q=2` even-scalar correction `z = 1.30913`,
+rank-2 Jordan log `z = 0.14675`; `results/server-20260829/P50-n145-n290-fullcurve/analysis/sprime_frozen_score.json`,
+`analysis/score.json` and REPORT.md table), with q=2 first in the declared order.
+The paper's mechanism claim stops exactly here: **low-rank non-scalar finite-size
+mixing, with a shape mode carrying the rejection; no module identification.**
+`docs/STATUS.md` line 68 records the `x = 21/4`-candidate reading of the leading sector;
+this draft quotes that level and does not exceed it.
+
+## 8. Limitations
+
+1. **Correlated-block accounting (§4) governs every "combined" reading of the results.**
+   The only multi-block claim is the two-block zero rejection (Block I+II). Blocks II and
+   III contributions are single-block statements; P48 shares its random block with P43's
+   primary score; the thermal-jet and central-contrast tables reuse primary raw curves.
+   GOVERNANCE §2E.
+2. **The N=580 aspect ladder does not adjudicate the amplitude law (#577).** Verdict
+   committed: `"underpowered: more than one competing prediction survives at r4_over_r1"`
+   (`results/aspect-ladder-n580/latest.json`, `verdict`; narrative
+   `notes/aspect-ladder-n580-result-20260905.md`). Three survivors — bare aspect ratio
+   (`4.00`, z `+0.50`), weight-4 modular shape (`10.99`, z `−2.08`), plain area scaling
+   (`16`, z `−2.56`) — one clean exclusion (`no_modulus_dependence`, z `+9.5`), 3σ
+   Fieller interval `[2.40, 27.47]`. The shape/modulus assignment of the `S'` partner is
+   therefore a limitation, not a finding, of this paper; the paper's exclusion claims must
+   not be read as selecting among the three survivors. (Cross-rung covariance measured by
+   replay: ρ `−0.1648`; moves no verdict — same artifact.)
+3. **Block II alone does not reject a zero child effect** (§5.2 clause). The rejection of
+   global zero rests on the synthesis of §5.1; alias selection and signal existence are
+   different claims with different blocks.
+4. **Prospective discipline is per-prediction, not per-paper.** The P48 pure law was
+   trained on the earlier sizes N=65/85/130 (a retrospective source) but frozen before
+   the N=185/265 target values existed (`predictions/p48_sprime_correction_20260828.yaml`,
+   `status: frozen_before_target_reveal`; P48 REPORT.md "Frozen source"); the falsification
+   is prospective against that new geometry, and §6.3 says so in those words. The
+   `2^(3/8)` baseline of §5.3 is a frozen asymptotic reference, not a rival fit.
+5. **No covariance pooling across blocks beyond the block-diagonal contract.** The §5.1
+   joint scores add two frozen covariances; the NLPD comparison is offered precisely
+   because the raw chi-square difference is *not* a valid LR statistic
+   (`notes/issue212-matching-odd-synthesis.md`).
+6. **Semantics are load-bearing.** The cross/either sign map is exact but channel- and
+   parity-specific (even: `−1`; odd: `+1`); an earlier interpretation error (#108) shows
+   the failure mode. The paper inherits the typed-gate requirement (source/target channel
+   fields, exact map) as reporting hygiene — erratum "Governance consequence".
+7. **The scalar-width failure is scored on a 10-dof full-covariance collapse**; the
+   ordered successors survive differently at their own levels (§6.2), and none is
+   promoted by this draft.
+8. **The three-level transfer rejection is at N=145→290, one lineage.** A second
+   independent doubling curve (new geometry, not a rerun) is the natural referee ask; we
+   note it rather than pre-empt it.
+
+## 9. Relation to `docs/STATUS.md` (read-only)
+
+Cited levels (as of `main`, this draft changes nothing): line 25 C3 (signal exists);
+line 27 C3 (H4 beats aliases); lines 29, 31 C3 negative (multiplier, pure `S'` law);
+line 30 C3 (frozen slope correction); line 32 C2 negative (scalar width); line 36 C3
+negative (weight-4 amplitude law across aspects — underpowered); line 68 candidate
+reading. Levels change on evidence, not on prose.
+
+## 10. What had to be left out (for the next draft)
+
+1. **Generated tables and a machine-checkable assembly.** A
+   `scripts/p4_manuscript_evidence_table.py` + `tables.md` + drift test in the `p2`/`p3`
+   style, so no number in the manuscript is typed. Until then §5–§7 carry explicit
+   `path → JSON field` citations instead.
+2. **A `references.bib` pass with source verification.** The current bib
+   (10 keys) has no finite-size-scaling or crossing-probability entries; the paper will
+   need at least Cardy/Watts crossing-probability and standard FSS references, each read
+   and marked per repo `[LIT]`/`PRIMARY_TEXT_READ` convention (p2's README documents that
+   workflow). Deliberately not added under a no-new-work ticket — invented bib entries
+   would violate the same discipline that makes the falsifications count.
+3. **The Q4/Jordan module bridge and the shape/modulus question**, which the ladder
+   limitation (§8.2) keeps open; also the #138 conjugation phase-node diagnostic
+   construction (P50 REPORT "no new Monte Carlo block is needed" note), which belongs to
+   a mechanism paper, not to this one.
+4. **A power analysis for the shape-mode rejection** (what curve count separates frozen
+   shape predictions at the current SE scale) — analysis, not computation, but new work.
+5. **Venue decision** — explicitly deferred by #644 and non-blocking.
+
+## 11. Evidence index
+
+| # | number | committed artifact (field) |
+|---|---|---|
+| 1 | `31.18573555150965 / 4`, `2.805595267905808e-6` | `results/evidence-ledger/issue212-matching-odd-synthesis.json` `joint_scores.zero_effect` |
+| 2 | `3.4622795373044295 / 4`, `0.48363695393249573` | same, `joint_scores.fixed_H4` |
+| 3 | NLPD delta `-13.806418789729096` | same, `predictive_comparison` |
+| 4 | `0.41630376401835489631 / 2`; `35.193078878051031476 / 2`; `16.012022412848537938 / 2`; `1.7764 / 2` | `results/server-20260829/P57-norm5-500m/primary_score.json` (models) |
+| 5 | H4/H12 frozen multipliers | `predictions/gaussian_norm5_harmonic_discrimination_20260828.yaml` (`exact_harmonic_predictions`) |
+| 6 | `9.352003684819664 / 2`, `0.00931618710954876`; eigenvalues; mode split `1.9243/7.4277` | `results/server-20260829/P50-n145-n290-fullcurve/analysis/score.json`; `REPORT.md` |
+| 7 | `z = -0.6660797543793089` vs `-22.690348726453408` | same, `slope.*` |
+| 8 | `24.500424925937486902 / 10` (`0.006377`); `22.2386 / 10`; `17.0513 / 10` | `results/server-20260829/P57-norm5-500m/thermal_jet_score.json` `primary_width_collapse.score`; `THERMAL_JET_SCORE.md` |
+| 9 | `52.71633588357711 / 2`; siblings `1.139/0.281/0.088`; `A0 = 1.9434247576878727` | `results/server-20260828/P48-new-geometry-score/score.json` `channels.*`; `REPORT.md` |
+| 10 | `0.5700315435551194 / 2` (`+0.6672, −0.1189`) | `results/server-20260828/P43-heldout-fullcurve-500m/analysis/channel_map_corrected_DeltaS.json`; `notes/issue43-cross-either-channel-map-erratum.md` |
+| 11 | Russo identity + pivotal bridge | `notes/pivotal-russo-finite-volume.md`; `scripts/exact_pivotal_russo.py`; `results/local-20260829/P100-pivotal-h4-bridge/score.json` |
+| 12 | ladder verdict; Fieller `[2.40, 27.47]`; survivors | `results/aspect-ladder-n580/latest.json` (`verdict`, compatible set); `notes/aspect-ladder-n580-result-20260905.md` |
+| 13 | P48 frozen correction targets (N290): q2 `1.30913`, Jordan `0.14675` | `results/server-20260829/P50-n145-n290-fullcurve/analysis/sprime_frozen_score.json`; `REPORT.md` |
+| 14 | D'/S' split `−0.00909` / `+2.69536` | `results/server-20260829/P50-n145-n290-fullcurve/REPORT.md` (Correlated P4 diagnostics) |
+| 15 | reproduction: `scripts/score_matching_odd_synthesis.py`, `scripts/score_norm5_harmonic_primary_typed.py` | `notes/issue212-matching-odd-synthesis.md`, `notes/norm5-harmonic-primary-scorer.md` |
diff --git a/docs/manuscripts/p5-exact-continuation-bounded-state/README.md b/docs/manuscripts/p5-exact-continuation-bounded-state/README.md
new file mode 100644
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+# P5 — What a bounded state can and cannot carry: exact continuation, branching, and a no-go
+
+Draft manuscript notes for **Paper 2** of [issue #644](https://github.com/LightChainr/Matching-One/issues/644).
+Portfolio track **P5**, following the `p2`/`p3`/`p4` manuscript convention.
+
+| File | Role |
+|---|---|
+| `README.md` (this file) | the draft: results, boundaries, limitations, omissions |
+
+**Status of this document.** A first-pass writing draft only. It promotes nothing,
+computes nothing, and touches `docs/STATUS.md` not at all; levels quoted below are current
+`main` content, cited as-is. Issue #644 stays open. No venue selected; the ticket defers
+that and this draft does not pretend otherwise. Unlike `p2`/`p3` there is **no generated
+`tables.md` and no evidence-assembly script yet** — every value below is carried with the
+committed file path (and field, for JSON) where its full-precision value lives. All paths
+are relative to the repository root.
+
+## 1. Abstract (draft)
+
+Three exact results about the limits of bounded state descriptions of continuation
+processes. (i) On a family of two-terminal cut networks, the complete unbranched survival
+law can be *identical* while branching behaviour separates `k+1` exact predictive classes,
+and the response-matrix rank achievable by experiments of depth `≤ d` is exactly
+`min(d+1, k+1)`: richer branching futures require predictive classes that no fixed depth —
+and no bounded-radius terminal summary — supplies. (ii) A constructive no-go: two analytic
+families of local, irreducible-at-finite-`L` Markov generators with a *common declared
+task* have identical finite-horizon responses to machine precision and closing-gap
+thresholds at `1/2` and `1/3` respectively; bounded finite-horizon task rank therefore
+carries no threshold information unless critical-sector completeness is separately
+assumed. (iii) For a finite connectivity process (P398), the coarsest admissible exact
+positive lumping is *provably equal, block for block*, to the orbit partition of the one
+reflection its declared readout dictionary preserves — a first-order response selection
+rule makes the symmetry-breaking direction invisible to the invariant task (the Duhamel
+integrand vanishes pointwise on protected pairs, while the deliberately exposed channel
+fires at odd widths) — and the task's balanced realization factors exactly through the
+quotient because the odd sector is inert. The through-line: compressibility, symmetry
+structure, and threshold content are separate properties of a state description, and
+conflating them is the failure mode each result closes. The reflection-fixed counts in
+(iii) are classical (Callan–Smiley; Burnside; Ding) and are cited, not claimed; the
+quotient identity is verified at five widths and stated as an instance, not a theorem —
+coarsest lumping need not equal an automorphism-orbit partition (D'Angeli–Donno,
+Prop. 13). P398 is a calibration process, not a percolation model; the result is a
+statement about the frozen finite-horizon input/output object.
+
+## 2. In its own voice: what this paper is not
+
+Per the ticket, this belongs in the body, not a footnote:
+
+- **P398 is not percolation.** It is a finite positive width-`w` process program used as a
+  calibration and mechanism-de-identification object (`docs/RESEARCH-ATLAS.md` §11). Part
+  III is an exact finite symmetry theorem about that declared input/output process. The
+  committed claim boundary already says so in the working-note voice —
+  `notes/p598-phase-c-balanced-quotient-20260906.md`: "this is a statement about the frozen
+  finite-horizon input/output description of one exactly known process, not about the state
+  space, and not a percolation result."
+- **The cut networks are proof devices in a declared gauge, not physical marks.** The
+  occupied-bridge convention is stated as such in
+  `notes/no-bounded-radius-quotient-20260906.md` §3 ("the network is a proof device in a
+  gauge, not a physical mark"), after #491/#550.
+- **No continuum claims.** Nothing here bounds the state dimension of a continuum field
+  theory, identifies an operator, or touches `p_c` numerically. Part II is precisely the
+  warning against reading threshold content out of state-counting.
+- **Levels.** `docs/STATUS.md` line 38 carries Part III's Phase C at level
+  "exact (bounded to the frozen I/O object)". Parts I and II have **no `docs/STATUS.md`
+  lines** on `main`; they are indexed in `docs/RESEARCH-ATLAS.md` §9 (integration
+  `open_pr` for #491/#492) and the #435/#549 rows of its integration table. This draft
+  reports that as-is and does not treat writing as promotion.
+
+## 3. Part I — Branching continuations need unbounded predictive classes (#435 / #491 / #549 / #550)
+
+**The fact that starts it.** Two N16 torus configurations with the *identical complete
+unbranched survival vector* `(1, 7/8, 9/14, 5/14, 4/35, 0, 0, 0, 0)` — one `L_0`
+predictive class — where a single declared fork separates them
+(`notes/memory-branching-and-moment-algebra-20260906.md`, FACT #435). The exact fork
+probability on the parallel-gadget family (#549) is affine,
+`F_{k,a} = [343k^3 − 182k^2 + 25k + 4a] / [8k(8k−1)^2]`, strictly increasing in the hidden
+class count `a` with consecutive gap `1/[2k(8k−1)^2]` — `1/98` at `k = 1`, reproducing
+#435's witness (`notes/probe-predictive-rank-separation-20260906.md`,
+`notes/branching-hankel-rank-lower-bound-20260906.md` §4).
+
+**Theorem A (rank upgrade).** Grouped-fork experiments `E_1..E_k` of depth `1..k` give a
+`(k+1)×(k+1)` response matrix of exact rank `k+1` for every `k`; machine-verified by exact
+rational elimination at `k = 2..8` — committed rank/class table in
+`results/cutnetwork-rank-vs-depth/latest.json` (keys `2..8`, each `rank = classes = k+1`);
+statement and proof sketch in `notes/branching-hankel-rank-lower-bound-20260906.md`
+(Theorems 1–2). Script `scripts/cutnetwork/rank_lower_bound.py` (note: the working note's
+Files section names `scripts/rank_lower_bound.py` / `results/rank-vs-depth/` — the
+committed locations differ and are the ones cited here).
+
+**Theorem B (depth is the resource).** The span of all experiments of depth `≤ d` has
+dimension `r_d(k) = d+1` for `d ≤ k`: the class count grows like `k+1` while every fixed
+depth sees a constant-size response space. The unbranched language (`d = 0`) sees
+essentially nothing; #549's single-fork language is exactly `r_1 = 2`.
+
+**Theorem C (no bounded radius suffices).** For every radius `r ≥ 0` and every `k`, `k+1`
+hidden classes admit planar two-terminal realizations sharing the unbranched survival law
+and the radius-`r` terminal neighbourhood, with fork gap `1/[2k(8k−1)²]` — the bank is
+pushed beyond the radius on an occupied bridge. Hence no bounded-radius quotient of the
+cut network preserves all branching futures.
+`notes/no-bounded-radius-quotient-20260906.md`; script
+`scripts/cutnetwork/radius_insufficiency.py`; data
+`results/cutnetwork-radius-insufficiency/latest.json` (radii `1,2,4,8`). This upgrades
+#550's `r = 1` certificate at every radius simultaneously, and already for depth 1.
+
+**Theorem D (ordinary rank vs nonnegative rank).** An explicit family (slack matrices of
+the regular `n`-gon) with ordinary rank `≤ 3` and nonnegative rank `n` (4 at `n = 4`) —
+so even positivity, the natural constraint on probability responses, does not rescue a
+bounded signed summary. `notes/positive-vs-signed-slack-separation-20260906.md`; script
+`scripts/cutnetwork/slack_family.py`; data
+`results/cutnetwork-positive-vs-signed-slack-family/latest.json`.
+
+**Background the paper must cite but not claim:** #491's reduction of embedded rank-one
+continuation to two-terminal vertex connectivity, with the repository's own boundary —
+"an exact finite continuation representation, not a proof of bounded continuum state
+dimension" — quoted from `docs/RESEARCH-ATLAS.md` §9.
+
+## 4. Part II — Bounded task rank carries no threshold content (#615)
+
+**Setting.** Declared task `(G_L(p), B_L, C_L)`, finite-horizon response
+`R_L(t,p) = C_L e^{tG_L(p)} B_L`; exact task order = Kalman controllable-observable
+dimension; closing-gap threshold defined analytically (gap positive and analytic at every
+finite `L`, vanishing in the `L → ∞` limit) — all in
+`notes/bounded-task-rank-threshold-no-go-20260907.md` §0.
+
+**Theorem E (no-go, constructive).** Two analytic Markov families
+`G_L^{(j)} = G_vis ⊕ H_L(p − p_c^{(j)})`, `j = 1,2`, share source/readout: visible sector
+a fixed 2-state chain; hidden sector a biased nearest-neighbour walk on `{1..L}` with
+reflecting boundaries, rates `a = e^{p−p_c}`, `b = e^{−(p−p_c)}`, exact gap
+`γ_L = a + b − 2√(ab)·cos(π/L) → 4·sinh²((p−p_c)/2)`. The hidden chain is connected,
+positive, local, irreducible at every finite `L`. The responses are *identical* for all
+`L, p, t`, while `p_c^{(1)} = 1/2 ≠ 1/3 = p_c^{(2)}`; both have `sup_L r_ex = 2`.
+
+Committed witness: `results/theory-no-go/latest.json`
+(`witness.family1_pc = 0.5`, `witness.family2_pc = 0.3333…`, per-`p` response vectors
+byte-identical between families and equal to the visible-only response); generator
+`scripts/theory/no_go_theorem.py`. (The note's Files section lists
+`scripts/no_go_theorem.py` / `results/no-go/latest.json`; the committed paths above are
+what this draft cites.)
+
+**Why the counterexample is not a cheat (and where it is genuinely limited).**
+Proposition 1 (hidden ⟺ off-Kalman-CO) is the certificate that the *missing hypothesis*
+is completeness, not block-diagonality. The paper then states the three-level honesty
+ladder from the note §3: (a) the direct sum; (b) symmetry-protected hiding, which is
+necessarily reducible (an irreducible generator admits only trivial commuting finite-group
+actions with >1 isotypic type) — P398's reflection-odd sector is the worked instance;
+(c) irreducible exact masking, which *is* fine-tuned: committed numbers in
+`results/theory-irreducible-masking/latest.json` (`partial_fraction_coeff`: generic
+`0.48371173070873924`; readout-masked `4.1291408707408205e-17`; source-masked
+`−2.312360816083912e-17`; both `−1.9739160635202627e-33`; script
+`scripts/theory/irreducible_masking.py`). What survives generically in the irreducible
+class is *weak coupling*, not an exact dark sector — the note says so and the paper keeps
+that qualifier.
+
+**Corollary / scope sentence (paper-ready, from the note §5):**
+
+> "A bounded finite-horizon balanced order of a declared task certifies the
+> compressibility of that task's input–output map and the existence of an exact or
+> symmetry-protected uncontrollable/unobservable sector; it carries no information about
+> any thermodynamic threshold unless the critical modes are separately shown to be
+> controllable and observable."
+
+**Escape conditions (note §4), stated as the theorem's other half:** critical-sector
+completeness is *necessary* and not sufficient; a horizon growing with the correlation
+length amplifies but cannot create coverage (exact order is horizon-independent by
+analyticity — Lemma 0); a signed-observable zero with a monotonicity argument (the
+repository's own `M_L = P_2 − P_0` route) is an *independent* mechanism that never goes
+through rank at all. Task compressibility ≠ threshold identifiability: statements about
+`(G,B,C)` vs about `spec(G_L)` plus a sign-change argument.
+
+## 5. Part III — The exact reflection quotient (#598 / #600 / #601)
+
+**The object.** P398's positive lumping at widths `w = 4..8` on noncrossing partitions.
+The reflection named by the *task*: `R = (i ↦ (w−1) − i)` — of the eight declared readouts,
+seven are fully dihedral-invariant and `wrap = [state[0] == state[w−1]]` is preserved
+exactly by `R`, the unique reflection preserving the declared `D0` dictionary at every
+width (`notes/p398-reflection-parity-20260906.md`).
+
+**Theorem F (quotient identification, at five widths — an instance, §7.2).** By exact
+partition refinement, the coarsest `D0`-admissible strong lumping (stable for the declared
+`J`, `D` moves) **equals the `R`-orbit partition, block for block** — not merely in
+cardinality — at `w = 4,5,6,7,8` with `10, 26, 76, 232, 750` blocks; and the test bites:
+at odd widths `halves_linked` is not `R`-invariant, so demanding all eight readouts
+collapses the coarsest admissible partition to the identity at `w = 5` (all 42 blocks).
+Reproduce: `scripts/p398_reflection_parity.py` →
+`results/p398-reflection-parity/latest.json` (fields `by_width[*]`, coarsest-vs-orbit
+equality; 17 tests in `tests/test_p398_reflection_parity.py`).
+
+**The counts are classical.** `[Catalan(w) + C(w,⌊w/2⌋)]/2 = 10,26,76,232,750` is Burnside
+for `{1,R}` once the reflection-fixed count `C(w,⌊w/2⌋)` is known; that count is
+Callan–Smiley (arXiv:math/0510447, Thm 1, 2005) with Ding 2016 (Miami PhD, Thm 2.1.2,
+Kreweras anti-isomorphism of the two even-`n` reflection classes) for the even-`n`
+second class; equivalently OEIS A007123(`w+1`). **The paper cites; it claims nothing here.**
+Full bibliographic detail, verbatim quotes and the explicit do-not-cite list (Reiner–
+Stanton–White cyclic sieving is the *rotation* count; type-B NC counts a different object)
+are in `notes/literature-officer-20260906-issue601.md` §Q1. The out-of-sample predictions
+`r_positive(9) = 2494`, `r_positive(10) = 8524` are stated in the notes (as
+A007123(10)/(11)); **widths 9–8…10 have not been run in this repository** — §8.2.
+
+**Theorem G (parity selection rule — pointwise).** Decompose the localized tilt into
+`R`-even and `R`-odd halves; over half the breaking direction is odd
+(`||H_odd||/||H_single|| = 0.55277, 0.55902, 0.56026, …` for `w = 4..8`; committed at
+`results/p398-reflection-parity/latest.json`, fields `by_width[*].parity_split.odd_share_of_norm`).
+For an invariant baseline and `R`-even source/readout, the first-order Duhamel integrand
+`μᵀ e^{(t−s)G} H_odd e^{sG} f` vanishes **pointwise**, not merely in integral — checked on
+a declared grid for all four sources against every readout: protected-pair worst relative
+integrand `~6.6e-15` or below; **positive control:** `halves_linked` is odd-containing at
+`w = 5, 7` and the rule *requires* it to fire there — observed largest exposed relative
+integrand `0.292092871260188` (w=5) and `0.10691283943577032` (w=7), same artifact
+(`by_width[*].selection_rule.largest_exposed_relative_integrand`). "A run in which
+everything vanished would have been a broken test, not a stronger theorem."
+Phase B (second order): the odd direction moves the invariant task at slope `2.0006–2.0010`
+in `ε`, the even direction at `0.98`, a factor 26 at `ε = 1/4`, `w = 8`
+(`notes/p398-reflection-parity-20260906.md` Phase B).
+
+**Theorem H (factorization).** The task's balanced realization factors through the
+quotient: microscopic vs quotient constructions agree with `max |A−B| ≤ 2.7e-15` in every
+resolvable Hankel direction at `w = 4..8`, balanced order and numerical rank preserved;
+the odd sector (`dim = 4/16/56/197/680`) is **exactly inert**, reach/observe energy
+`1.69e-17 … 5.69e-17` — which is why the reduction is exact, not merely accurate. The
+declared trap fires: the exposed dictionary breaks at exactly `w = 5, 7`
+(`0.017944…` / `0.008650…`) and only there, recovering to `1e-16` under symmetrization.
+`notes/p598-phase-c-balanced-quotient-20260906.md`;
+`scripts/p598_phase_c_balanced_quotient.py` → `results/p598-balanced-quotient/latest.json`
+(fields `blocks[*].dictionaries.*`, `odd_sector.reach_odd_energy_relative`,
+`decision.*`); `tests/test_p598_phase_c_balanced_quotient.py`. STATUS level: line 38,
+"exact (bounded to the frozen I/O object)" — the parenthetical is the claim ceiling and the
+paper keeps it.
+
+**Prior-art position (#601).** Round 1 and round 2 of the literature pass
+(`notes/literature-officer-20260906-issue601.md`,
+`notes/literature-officer-20260906-issue601-round2.md`) establish: the abstract vanishing
+is Schur / Wigner–Eckart; the *statement* for equivariant Markov generators with the
+Duhamel integrand vanishing pointwise "is still unnamed" (Hänggi I/II,
+Golubitsky–Stewart–Schaeffer, Antown–Dragičević–Froyland 2018, Diaconis 1988 Ch. 3E,
+Santos Gutiérrez–Zagli–Carigi 2025 all checked and are adjacent, not it). The paper's
+contribution there is **language and the pointwise verification**, claimed as a remark
+citing Schur, not a theorem. Phase H's factorization is the Kalman decomposition applied
+to a `C2`-equivariant realization — cite Wonham, per round 2's own instruction; the
+odd-width exposed control is what makes the test non-vacuous.
+
+## 6. The through-line
+
+1. **What a bounded state carries about *futures*** (Part I): unbranched memory can be
+   complete and still blind; branching depth, not width, is the resource; positivity of
+   responses does not help (Thm D).
+2. **What a bounded task carries about *thresholds*** (Part II): nothing, without
+   critical-sector completeness; the witness is analytic, local, irreducible at finite
+   size, with the response identical between families to machine precision.
+3. **What a bounded state carries about *symmetry*** (Part III): exact quotients *are*
+   automorphism orbits when they exist — and here the coarsest lumping provably equals one,
+   while first-order response selection rules say precisely which symmetry-breaking
+   structure any such description is blind to, and the factorization theorem says what
+   survives the blindness intact.
+
+Together: compressibility, branching expressiveness, threshold content and symmetry
+structure are four separate properties. Each part is a theorem or construction about one
+implication failing to run to the others.
+
+## 7. Boundaries the prose must respect (settled by the ticket; restated here)
+
+1. Reflection-fixed counts: **cite, do not claim** (Callan–Smiley Thm 1 + Burnside; Ding
+   2016 even-`n`; OEIS A007123). Cardinality matching identifies nothing — every
+   reflection fixes `C(w,⌊w/2⌋)`; the load is carried by the partition equality and the
+   task that names `R` (already correct in the working note).
+2. "Coarsest lumping = automorphism-orbit partition" is **not a theorem in general**
+   (D'Angeli–Donno 2013, Prop. 13 gives a lumping induced by no subgroup; their Thm 12 is
+   the version *with hypotheses*, which P398 does not satisfy). The `w = 4..8` agreement
+   is **an instance** and the abstract above calls it one. Gate 1's branch-A resolution
+   must never be written as "by the theorem relating coarsest lumpings to Aut orbits."
+3. P398 is a calibration process, not percolation — §2 in the paper's own voice.
+4. Parts I and II: the constructions are about the *declared* protocol (the published
+   #549 fork composition, the occupied-bridge gauge, the declared `(G,B,C)` triples); the
+   full-N16 lift is explicitly not claimed (`notes/branching-hankel-rank-lower-bound-20260906.md` §5).
+5. #636 territory and any `p_c` inference are out of scope entirely.
+
+## 8. Limitations
+
+1. **Five widths.** The quotient identification is proved by exhaustive exact refinement
+   at `w = 4..8` only; it is a finite instance, and the generic-rates lemma that would
+   turn it into a theorem with hypotheses has *not* been written (round 2 checked the
+   neighbourhood — Watanabe–Wolfer 2024/2026 is adjacent and does not contain it).
+2. **Widths 9–10 untested.** `r_positive(9) = 2494`, `r_positive(10) = 8524` are
+   predictions against A007123; the partition equality re-derivation commits at
+   `w = 4..8`; no committed artifact in this repository extends it (or the cardinality) to
+   widths 9–10. Reportable as a finding
+   rather than produced: per this ticket's rule the gap is stated, not computed.
+3. **Part I's theorems are family statements.** They bound what the declared fork/radius
+   languages can carry; they do not exhibit a percolation observable whose branching
+   futures *require* unbounded classes at any fixed depth (the gap from proof device to
+   lattice observable is open and the note's §5 says so).
+4. **Part II's strongest irreducible form is fine-tuned or weak-coupling** (§4c): exact
+   hiding in the irreducible/local class survives only as single-mode orthogonality; a
+   generic irreducible task has only `O(ε)` hiding. The no-go's force is for *declared
+   tasks and symmetry-protected sectors*, which is what the paper claims.
+5. **Phase C's agreement is bounded to the frozen I/O object** (STATUS line 38
+   parenthetical): the `Catalan − r_reflect` "extra" states are not asserted unphysical.
+6. **The #601 gap is a negative literature result**, bounded to the searches recorded in
+   the two packets; a selection-rule statement could exist under language neither pass
+   indexed.
+7. **No STATUS promotion.** Parts I–II have no claim-ledger rows; that is the honest state
+   of integration (`open_pr` snapshot per atlas), and the draft treats it as such.
+
+## 9. What had to be left out (for the next draft)
+
+1. `references.bib` entries for this cluster — Callan–Smiley 2005, Ding 2016,
+   D'Angeli–Donno 2013, Kemeny–Snell 1960, Buchholz 1994, Godsil–Royle 2001,
+   Hänggi 1978 I/II, Wigner–Eckart (textbook cite), Diaconis 1988, Antown–Dragičević–Froyland
+   2018, Wonham, OEIS A007123 — **are not in `references.bib`** (10 keys, none matching;
+   a finding). Deliberately not added here: the full bibliographic detail lives in the two
+   #601 packets and the bib patch should be their transcription with `PRIMARY_TEXT_READ`
+   marking per convention, done in a venue-prep pass, not silently inside a no-compute
+   writing PR.
+2. Generated tables / evidence-assembly script (`p2`/`p3` style) locking every Part III
+   number to its JSON field with a drift test.
+3. Widths 9–10 exact refinement run (new computation — separate ticket).
+4. The generic-rates lemma (§8.1) — optional theory that would upgrade instance→theorem.
+5. Related-work prose on lumpability/bisimulation (round 1 §Q2 "Bisimulation / coalgebra")
+   and the model-reduction Krylov-prefix material (round 1 §Q5) — adjacent to Part III/II
+   framing but beyond this paper's claim budget.
+6. Venue. Ticket says the owner picks once drafts exist; this draft neither proposes nor
+   pretends one was chosen.
+
+## 10. Evidence index
+
+| # | statement / value | committed artifact (field) |
+|---|---|---|
+| 1 | #435 identical unbranched vector; single `L_0` class | `notes/memory-branching-and-moment-algebra-20260906.md` (Program H, FACT) |
+| 2 | `F_{k,a}` closed form; gap `1/[2k(8k−1)²]`; `1/98` | `notes/probe-predictive-rank-separation-20260906.md`; `results/cutnetwork-radius-insufficiency/latest.json` |
+| 3 | rank `= classes = k+1` at `k = 2..8` | `results/cutnetwork-rank-vs-depth/latest.json`; `scripts/cutnetwork/rank_lower_bound.py` |
+| 4 | no bounded-radius quotient (all `r`) | `notes/no-bounded-radius-quotient-20260906.md` §2; `scripts/cutnetwork/radius_insufficiency.py` |
+| 5 | ordinary rank ≤ 3 / nonnegative rank `n` | `notes/positive-vs-signed-slack-separation-20260906.md`; `results/cutnetwork-positive-vs-signed-slack-family/latest.json` |
+| 6 | #491 cut-network theorem + boundary | `docs/RESEARCH-ATLAS.md` §9 |
+| 7 | Theorem E families; `1/2`, `1/3`; identical responses | `notes/bounded-task-rank-threshold-no-go-20260907.md` §2; `results/theory-no-go/latest.json` (`witness.*`); `scripts/theory/no_go_theorem.py` |
+| 8 | masking coefficients `0.4837…/4.13e-17/−2.31e-17/−1.97e-33` | `results/theory-irreducible-masking/latest.json` (`partial_fraction_coeff`); `scripts/theory/irreducible_masking.py` |
+| 9 | scope sentence (paper-ready) | note §5 quoted verbatim above |
+| 10 | `R` unique to dictionary; block-for-block equality; `10,26,76,232,750`; `w=5` identity collapse | `notes/p398-reflection-parity-20260906.md`; `results/p398-reflection-parity/latest.json`; `scripts/p398_reflection_parity.py` |
+| 11 | classical count citation + do-not-cite list; A007123; 2494/8524 | `notes/literature-officer-20260906-issue601.md` §Q1 |
+| 12 | odd share `0.55277…`; exposed `0.29209…/0.10691…`; protected `~1e-15` | `results/p398-reflection-parity/latest.json` (`parity_split.odd_share_of_norm`, `selection_rule.largest_exposed_relative_integrand`) |
+| 13 | Phase B slopes `2.0006–2.0010` vs `0.98`; factor 26 | note Phase B table; same artifact |
+| 14 | Phase C `≤ 2.7e-15`; inert odd `1.7e-17…5.7e-17`, dims `4/16/56/197/680`; breaks `1.8e-2/8.7e-3` | `results/p598-balanced-quotient/latest.json`; `scripts/p598_phase_c_balanced_quotient.py`; `tests/test_p598_phase_c_balanced_quotient.py` |
+| 15 | STATUS line 38 level | `docs/STATUS.md:38` (read-only) |
+| 16 | #601 gap statements (Q3 round 1/2) | `notes/literature-officer-20260906-issue601.md`, `...-round2.md` |