Price a second central coordinate on the selected p16 controller: it must be repaid; incremental realizations lose, allowance 1,859 (no new κ)

research/two-central-laurent-coverage/PROOF.md

@@ -0,0 +1,172 @@
+# Two-central substitution on the selected p16 controller: written argument
+
+JoyboyBrian, with substantial assistance from Anthropic Claude. Apache-2.0.
+
+## 0. Object and premises
+
+The object is the selected complete invoice of #4470
+(`research/joint-helper-lattice/RESULT.json`, sha256 `c7b2388f…5236c`, which is the
+selected record's `source_certificate_sha256`). Write `T` for its productive (target) rank
+histogram and `P` for its paid histogram. For `t = 1 - C` put `T_t = Σ T_k k^t`,
+`P_t = Σ P_r r^t` and `S(C) = T_t - P_t`. The selected fixed-invoice root lies in
+`(0.0030201659770, 0.0030201659771)`.
+
+Inherited, not re-proved: the native all-column semantics, the PR4440 Laurent controller
+factorization, the whole-job matcher, completion/splitting, compiler and outer transfer
+of the selected review. Nothing here changes the selected record.
+
+## 1. Where the single unpaid coordinate comes from
+
+`verify.py` checks on the pinned bytes:
+
+- The main targets are 224 of rank 145 = 1 + 8·18, 448 of each rank 1 + 8k for
+  k = 2,…,8, and 224 of rank 73 = 1 + 8·9. That is 2·V·p = 3,584 main targets, with
+  V = 112 ports and p = 16 pairs.
+- Every arm target 1 + 8k has a paid whole-idle of rank exactly 8k with the same
+  multiplicity. The long targets are covered by the 18 native calls.
+- The rank-one surplus is `752 = 18·(H_1 - P_1 + V + 2VB) - V`, with H_1 = 3,471,
+  P_1 = 5,327 and B = 8.
+
+The extra coordinate in each main target is the central Gaussian line `Q = C_q`, with
+`q ∈ F2^9` the port's odd vector (`kappa-hybrid-controller/NATIVE.md` §2). The Laurent
+controller has diagonal targets `D·Q^{±1}` (central exponent ±1, PR4470 PROOF §4). The
+paid idles carry only the D blocks. In this interface the old #1094 coverage identity
+`mass = 72W + 64N` and `ceiling.ceiling(unpaid=…)` do not apply. The replacement
+identity is the one above: one central credit per main target.
+
+## 2. The changed contract: substitution Q → Q_a Q_b
+
+Let `q_b` be a fresh spectator axis of every main field and `Q_b = C_{q_b}`. The PR4440
+factorization is an identity in the commutative Laurent ring generated by `Q, D_0, …`.
+The map `Q ↦ Q_a Q_b` (with `Q_a = C_q`) is a ring homomorphism into a commutative
+algebra, because operators on disjoint axes commute. So the substituted word is again an
+exact controller, and every main target becomes `D·Q_a^{±1} Q_b^{±1}`. These are **two
+unpaid coordinates per main family**, the card's object.
+
+`C_{q_a} C_{q_b}` is a rank-2 Gaussian request (kernel support spans {q_a, q_b}). Its
+inverse is `X_{q_a} X_{q_b} C_{q_a} C_{q_b}`, because `C_q^{-1} = X_q C_q`.
+
+The fully charged realizations considered are:
+
+1. **Thin shears.** The 15 query shears per port become rank-2 requests (+1 rank each).
+2. **Native contacts by conjugation.** With `Δ = diag(1, Q_b)` on the stage's (X, Y)
+   pair,
+   `D_s S_y(Q_a^{-1} Q_b^{-1}) = Δ^{-1} (D_s S_y(Q_a^{-1})) Δ` and
+   `S_x(-Q_a Q_b) = Δ^{-1} S_x(-Q_a) Δ`.
+   So each unchanged native call plus two rank-one `q_b` requests per port realizes the
+   substituted native stage.
+3. **Optimistic relaxation.** Item 1 with native calls unchanged and free. This is a
+   screen, not a realization.
+
+## 3. Credit-rank lemma
+
+Work over the field `K = F(x)[Q_b]/(Q_b^2 - x)`. Here F is the rational function field
+in all other symbols, and x stands for the translation `X_{q_b}`. Let `σ` be the
+Galois automorphism `Q_b ↦ -Q_b`. It fixes x, so it fixes `X_{q_b} = Q_b^2` and every
+`Q_b`-free operator.
+
+**Lemma.** Let `W = F_1 ⋯ F_m` be a program of square matrices over K. Every column is
+a physical field, including helpers and copies. Suppose its productive block, between
+`σ`-fixed boundary programs, is monomial with entries `u_i Q_b^{e_i}`, where each `u_i` is
+`σ`-fixed. Then
+
+    #{i : e_i odd} ≤ Σ_j rank(σ(F_j) - F_j).
+
+*Proof.* `σ(W) - W = Σ_j σ(F_1)⋯σ(F_{j-1}) (σ(F_j) - F_j) F_{j+1}⋯F_m`, so its rank is at
+most the right side. A submatrix and a product with fixed invertible boundary matrices do
+not increase rank. For the monomial target, `σ(T) - T = Π·diag(((-1)^{e_i} - 1) u_i Q_b^{e_i})`,
+whose rank is the number of odd `e_i`. ∎
+
+**Premise (P).** A factor is `σ`-fixed unless it is a paid Gaussian request on one field
+whose effective active span contains `q_b`. Scalar operations, copies, adds, routes fixing
+`q_b` and the translation `X_{q_b}` are `σ`-fixed. A request on one field is a diagonal
+factor, so `rank(σ(F) - F) ≤ 1`. A request inside a native call is counted
+individually; the native is a product of its operations.
+
+Under (P), **c credited targets require at least c paid requests containing q_b**. In the
+substitution contract, c = 3,584. The 15·112 = 1,680 thin shears supply at most 1,680, so
+the native stages must contain at least **1,904 q_b-touching requests** (≈105.8 per
+native call). The formal and literal instances in `verify.py` show both cases: the
+substituted three-shear word has 3 terms for 2 credits; the gauge realization has 2 terms
+for 2 credits, so the bound is tight there. A pure gauge on a diagonal target yields zero
+credits.
+
+## 4. Obstruction for incremental full-credit realizations
+
+**Incremental premise (I).** Relative to the selected invoice, the realization keeps
+every productive field. It adds `q_b` to targets only, so each main target R becomes R + 1.
+It keeps every selected paid request. Each `q_b`-touching request is either an existing
+selected request enlarged by `q_b` (rank r → r + 1, at most once per request) or an
+additional request of rank ≥ 1. Further costs only add paid work.
+
+Put `g(x) = (x+1)^t - x^t = ∫_x^{x+1} t y^{t-1} dy`. For `0 < t < 1` it is strictly
+decreasing, with `g(0) = 1`. An additional request of rank r' costs `r'^t ≥ 1 ≥ g(x)` for
+every x ≥ 0.
+
+**Theorem.** Under (P) and (I), if every one of the 3,584 main targets is credited (two
+unpaid coordinates per live main family), then `S_new(C) < S(C)` for every `0 < C < 1`.
+At C = 0 the gain is exactly repaid at best.
+
+*Proof.* The gain is `Σ_main g(R_i)`. By the lemma, at least 3,584 distinct q_b-touching
+requests are charged. Their cost is at least the sum of g over the 3,584 largest selected
+paid ranks; additional requests cost at least 1. `verify.py` checks exactly that the
+sorted main targets dominate these paid ranks elementwise. The pairs are
+145↔72, 73↔64, 65↔64, …, 17↔8 or 5, and every target is larger. Since g is strictly
+decreasing, the gain is strictly below the cost. ∎
+
+So the fixed-invoice root of any such realization is strictly below the selected root.
+The selected package certifies that `S` decreases through C = 1/20 and is negative beyond
+it; this is inherited. `verify.py` also encloses the net change at three points
+(selected root: ≤ −2.285).
+
+The conjugation realization is one instance: its rank-one surplus is
+`752 + 3584 - 1680 - 4032 = -1376`. It is negative already at C = 0.
+
+## 5. Partial credits: an explicit ceiling
+
+When only a subset of main families gets the second coordinate, dominance can fail. Small
+targets (17) can be paired with large idles (72). The verifier rejects that case as a
+mutation control. For any credited subset of main targets, of size c,
+
+    S_new(C) ≤ S(C) + max_c [ Σ_{c smallest main targets} g(R) - Σ_{c largest paid ranks} g(r) ].
+
+The second term is computed with outward intervals as a best prefix; 1,568 pairs have
+R < r. At the selected root it is at most 3.67. Bisection on a 1e-9 grid certifies that the
+right side is negative at **C = 0.003035069**. No partial-credit incremental
+realization certifies that saving, a ceiling of +0.49% over the selected root.
+
+This is a pointwise certificate at the stated point; the bisection's other points are
+sign checks. Credits on helper targets are outside this scope: they change helper
+residual frames, which is the helper-retiming mechanism, not a coverage law.
+
+## 6. What is left open: the native allowance
+
+In the optimistic relaxation (§2.3) the root is enclosed in
+`(0.01023582, 0.010235821)`, 3.39× the selected root. So the relaxation passes, and
+only the native charge decides the contract. Adding E rank-one paid units keeps a
+positive surplus at the selected root's upper end exactly for **E ≤ 1,859**
+(103.28 per native call; E = 1,860 fails). The lemma forces at least 1,904 q_b-touching
+requests in the native stages. Paying each one as a separate rank-one request
+therefore loses; the conjugation contract pays 4,032.
+
+By §4, enlarging the selected native requests does not help either. **The only
+route left open under (P) is a redesigned native word on the enlarged central block.**
+Its own complete local profile must change, and its moment must beat the selected
+`H_t - P_t` after it realizes ≥ 1,904/18 central-q_b contacts per call. No such word
+is constructed here.
+
+## 7. Small complete instance
+
+`verify.py` replays:
+
+- **Formal (Laurent polynomials in Q_a, Q_b, D).** The one- and two-central three-shear
+  identities; the native and back-shear conjugations; the gauge realization; the
+  credit-rank counts; and the diagonal-target zero-credit case.
+- **Literal (Gaussian-rational kernels on F2^3, axes q_a, q_b, d).** `C_q^2 = X_q`,
+  `C_q^{-1} = X_q C_q` and Fourier ranks 1 and 2. It runs six arbitrary dirty input pairs
+  (X, Y) through the substituted swap word and through the conjugated native-shaped
+  factor, and compares them with the declared targets.
+
+The literal algebra has `C_q ∈ span{1, X_q}`, so `σ` is not an automorphism there. The
+lemma is a statement about the formal Laurent model in which the controller is
+factorized; this is recorded as a modelling premise.

research/two-central-laurent-coverage/README.md

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+# A second central coordinate on the selected p16 controller must be repaid
+
+**This is a scoped obstruction with an explicit allowance. It claims no new κ and leaves
+the selected record unchanged.**
+
+Original card: `coverage-law-change` ("Can a template or call contract leave two or more
+unpaid coordinates per live family?"). Scope status is **refined**. The card's
+`72W + 64N` identity and `ceiling.ceiling(unpaid=…)` belong to the obsolete #1094
+template. This package rebinds the question to the selected #4470 invoice and derives a
+replacement bound; [TASK.json](TASK.json) gives the reason. The written argument is in
+[PROOF.md](PROOF.md). The exact certificate is [RESULT.json](RESULT.json).
+
+## Where the one unpaid coordinate comes from
+
+The selected invoice is #4470, Cinna Davis's p16 query-thinned controller, pinned
+byte-for-byte at `research/joint-helper-lattice/RESULT.json`. It has 3,584 = 2·112·16 main
+targets:
+
+- each arm target has rank `1 + 8k` and is paid by a whole-idle of rank `8k`;
+- each long target has rank `145 = 1 + 8·18`.
+
+The extra coordinate is the central Gaussian line `Q = C_q`, which enters every diagonal
+target with exponent ±1. The replacement coverage identity is exact:
+
+    rank-one surplus 752 = 18·(H1 − P1 + V + 2VB) − V   (V = 112, B = 8).
+
+## The changed contract and its price
+
+**Contract.** Substitute `Q → Q_a Q_b`, with `Q_b = C_{q_b}` on a fresh axis. Because all
+the symbols commute, this is a ring homomorphism of the Laurent factorization. Every main
+target becomes `D·Q_a^{±1}Q_b^{±1}`, so each live main family keeps **two** unpaid
+coordinates. The 15 thin shears per port become rank-2 requests. Native stages are realized
+exactly by the conjugation
+
+    D_s S_y(Q_a⁻¹Q_b⁻¹) = Δ⁻¹ (D_s S_y(Q_a⁻¹)) Δ,  Δ = diag(1, Q_b).
+
+| Profile (exact intervals) | Result |
+|---|---|
+| Selected invoice | root in (0.0030201659770, 0.0030201659771), reproduced |
+| Optimistic: natives unchanged and free, thin shears rank 2 | root in (0.01023582, 0.010235821), **3.39×** |
+| Conjugation-charged natives (+2 rank-one q_b requests per port per native stage) | rank-one surplus **−1,376**: no saving |
+| Native allowance over the optimistic profile at the selected root | **≤ 1,859** extra rank-one units (103.28 per call); 1,860 fails |
+| q_b-touching requests the natives must contain (credit-rank lemma) | **≥ 1,904** (3,584 credits − 1,680 thin shears) |
+| Any incremental full-credit realization (selected paid profile kept, q_b added to requests) | surplus **strictly below the selected surplus at every 0 < C < 1** |
+| Any incremental partial-credit realization (main targets only) | cannot certify **C = 0.003035069** (+0.49%) |
+
+**Credit-rank lemma** (PROOF §3, formal Laurent model). Let σ map `Q_b` to `−Q_b`. The
+number of targets with an odd `Q_b` exponent is at most `Σ rank(σ(F) − F)` over the
+program's factors. Each paid request containing `q_b` on one field contributes at most 1.
+So every second-coordinate credit needs its own `q_b`-touching paid request.
+
+**Dominance.** The sorted main targets dominate the 3,584 largest paid ranks elementwise.
+The marginal `g(x) = (x+1)^t − x^t` strictly decreases. So for every family, adding `q_b`
+to existing requests costs more than it earns.
+
+## What this establishes and what it does not
+
+- **Established**, under the premises (P) and (I) of PROOF.md:
+  - the second unpaid coordinate is exactly repaid at C = 0 at best, and is a strict loss
+    for every C > 0 when every main family is credited;
+  - partial crediting gains at most +0.49%;
+  - separate rank-one payment of the forced native terms fails, since 1,904 > 1,859.
+- **Not established:** a redesigned native word on the enlarged central block whose own
+  profile changes. That is the one route left open (see below).
+- Scope limits:
+  - The lemma lives in the formal Laurent model where the controller is factorized. In
+    the literal group algebra, `C_q ∈ span{1, X_q}`.
+  - Helper-frame credits are excluded.
+  - Native/all-column, compiler, completion, precision and outer premises are inherited
+    unchanged.
+- The small complete instance is checked two ways:
+  - formally, in Laurent polynomials;
+  - literally, with Gaussian-rational kernels on F2^3 and six arbitrary dirty input
+    pairs.
+
+  It covers the substitution and conjugation identities only. It is not a replay of the
+  p16 native word.
+
+## Reproduce
+
+Python 3.12, standard library only, about 8 seconds:
+
+```sh
+python3 scripts/research_package.py --package research/two-central-laurent-coverage --check-only
+python3.12 -B scripts/research_package.py --package research/two-central-laurent-coverage
+python3.12 -B research/two-central-laurent-coverage/verify.py        # PASS_TWO_CENTRAL_LAURENT_COVERAGE
+```
+
+`--write` regenerates RESULT.json. Otherwise the recomputed certificate must match it
+byte-for-byte.
+
+The verifier rejects six mutation controls:
+
+- an inverse without its `X_q` translation;
+- an uncharged second coordinate;
+- dominance claimed for small targets only;
+- allowance + 1;
+- a mutated main target;
+- a gauge on the wrong field.
+
+## Open next questions
+
+1. **A ten-axis native word.** Is there a redesigned native word on the enlarged central
+   block whose complete invoice realizes its ≥ 1,904/18 `q_b` contacts per call? It must
+   still beat the selected native's `H_t − P_t`.
+   - Why it matters: this is the only route the lemma and the incremental theorem leave
+     open, and it is worth up to 3.39×.
+   - First step: lift one fresh-helper stage to a ten-dimensional frame lattice and price
+     the `Q_aQ_b` cross term against the 1,859-unit allowance.
+   - Blocker: there is no ten-axis contact proof.
+2. **Helper-frame credits.** Can the second coordinate be credited to helper exits
+   (rank ≤ 9) instead of main targets?
+   - This belongs to the helper-retiming mechanism and is excluded here.
+   - First step: count admissible `q_b` helper exits under premise (P).
+3. **A global partial-credit ceiling.** Can the pointwise certificate at C = 0.003035069
+   become an all-C ceiling?
+   - First step: bound each paired term's C-derivative by `ln((r+1)/R)`.
+   - Not blocked.
+
+## Credit
+
+- Selected construction and p16 controller: Cinna Davis (1337isnot1337), #4470 and
+  #4440, with the credited native, helper-retiming and outer lineage in the selected
+  review.
+- Laurent shear and conjugation conventions: `research/kappa-hybrid-controller/NATIVE.md`.
+- Complete-surplus model: `research/controller-stage-ceiling` (Cinna Davis).
+- Exact ln/exp enclosures: Douglas Colkitt's `research/complex-local-word-ceiling/ceiling.py`
+  (#1240), reused unchanged.
+- Earlier coverage-law work on the #1094 template, which this package does not reprice:
+  - #1012's coverage identity;
+  - Navinor's `complex-coverage-law-relief`;
+  - Clear1nsight's `selected-family-coverage-boundary`;
+  - `three-bank-restored-unpaid` and `line-port-unpaid-cap`.
+- Distinct from the active centre-path, coded p16, two-FULL, H3-prefix and raw-dual lines
+  (Clear1nsight #4902/#4931/#4932/#4943/#4945; dillona #4930/#4933).
+
+Prepared by JoyboyBrian with substantial assistance from Anthropic Claude (Claude Code,
+Opus 5.5). Apache-2.0.

research/two-central-laurent-coverage/RESULT.json

@@ -0,0 +1,55 @@
+{
+  "base_commit": "cef4b52f2a59009865bf8eabde1be3f159063703",
+  "controls": {
+    "allowance_holds": true,
+    "allowance_plus_one_fails": true,
+    "inverse_without_translation": true,
+    "mutated_main_target_rejected": true,
+    "partial_credit_breaks_dominance": true,
+    "uncharged_second_coordinate": true,
+    "wrong_gauge_orientation_rejected": true
+  },
+  "coverage_identity": {
+    "helper_mass": 3471,
+    "identity": "surplus(C=0) = 18*(H1 - P1 + V + 2VB) - V = 752",
+    "main_targets": 3584,
+    "native_paid_mass": 5327,
+    "rank_one_surplus": 752
+  },
+  "formal_instance": {
+    "credits": 2,
+    "gauge_qb_terms": 2,
+    "interleaved_monomial": true,
+    "substitution_qb_terms": 3
+  },
+  "literal_instance": {
+    "central_ranks": [
+      1,
+      2
+    ],
+    "columns_replayed": 6,
+    "field_dimension": 3
+  },
+  "moments": {
+    "allowance_per_native_call": "1859/18",
+    "allowance_rank_one_units": 1859,
+    "conjugation_rank_one_surplus": -1376,
+    "full_credit_net_upper_at_selected_root": -2.2852989943453053,
+    "native_qb_terms_needed": 1904,
+    "native_terms_fit_allowance_as_rank_one": false,
+    "optimistic_ratio_vs_selected": 3.3891581050881707,
+    "optimistic_root": [
+      "511791/50000000",
+      "10235821/1000000000"
+    ],
+    "partial_best_net_gain_at_selected_root": 3.668156940596911,
+    "partial_credit_ceiling": "3035069/1000000000",
+    "partial_pairs_with_target_below_charge": 1568,
+    "selected_root": [
+      "3020165977/1000000000000",
+      "30201659771/10000000000000"
+    ]
+  },
+  "selected_source_sha256": "c7b2388f6511c19976b337d8c8ac374610af616866a249d75dace704eb5d9e21",
+  "status": "PASS_TWO_CENTRAL_LAURENT_COVERAGE"
+}