Prove intrinsic P104 translated-progress bound and audit canonical adapters

research/p104-translated-progress/PROOF.md

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+# Symplectic width, translated progress and the P104 adapter boundary
+
+## 1. Question and exact scope
+
+PR3954 and PR3962 condition their two-axis cost screens on at most 2312 units
+of persistent progress per unchanged P104 call. This study supplies a proof
+for a named, broad class of **matched translations of the intrinsic endpoints**,
+and an actual canonical-frame counterexample showing why the intrinsic
+endpoint qualification cannot be omitted. It does not prove that every
+translated program retains the old paid profile, preserves the caller's
+initial credit, or implements a new supplier.
+
+The argument combines the binary symplectic description of Clifford address
+operators with elementary rank-distance inequalities. These are standard
+linear-algebra tools; the source-bound endpoint application and adapter
+counterexample are the contribution. All matrix ranks in this note are over
+F2. They are not rational operator ranks on arrays, tensor rank, or arbitrary
+scalar-operation lower bounds.
+
+## 2. The native width and a progress potential
+
+Use Pauli coordinates (x,z), with X^x Z^z, and the unsigned symplectic matrix
+
+    M = [A B; C D],       phi(M)=rank(B).
+
+The pinned native compiler `vendor/clifford_exact.py:lower_child` sets its
+positive child width to the rank of the X parts of the images of the Z
+Paulis, exactly phi. Its `monomial` class has B=0. A child C on r independent
+address bits has B of rank r. Binary address permutations, quadratic
+fourth-root diagonal phases, Pauli translations and scalar phases have B=0.
+Their physical work is still paid; saying phi=0 does not make them free.
+The source compiler supplies a one-child normal form. For the bound below,
+only the lower-bound direction is needed: any factorization into these
+monomials and rank-r children has sum r >= phi of its product.
+
+Indeed,
+
+    B(MN)=A(M) B(N)+B(M) D(N),
+    phi(MN)<=phi(M)+phi(N).
+
+For a symplectic matrix, M^-1=[D^t B^t; C^t A^t] in characteristic two, so
+phi(M^-1)=phi(M). Thus
+
+    |phi(M)-phi(N)| <= phi(M N^-1).
+
+One can instead use N^-1 M by factoring on the other side. A monomial factor
+on either side preserves phi exactly: its relevant A or D block is
+invertible. In contrast, arbitrary Clifford conjugation need not preserve
+phi. Conjugating an X rotation by a Hadamard turns it into a diagonal Z
+rotation, changing positive width from one to zero. The Hadamard and any
+other inserted transformations require their own invoice.
+
+For a program with a separate invertible Clifford chart Gamma_r on each
+persistent physical role, use Phi=sum_r phi(Gamma_r). Native transports pay
+at least each signed increment of Phi. Scalar additions in an identical
+chart do not change charts or Phi, regardless of the logical row coefficients.
+A role permutation preserves the sum, but its realization/return is not
+thereby free. Summing the signed increments telescopes; decreases cannot be
+credited twice. Remaining work made from this named grammar therefore pays
+at least max(0,Phi_final-Phi_current). For the required full tensor endpoint,
+phi(F_m)=m on each original role. This proves the relevant native-progress
+inequality when the role-local/common-chart invariant actually holds.
+
+This statement does not cover a general matrix of address operators coupling
+several physical roles, a non-Clifford scalar algorithm, unproved owner
+relabeling, or changed initial/final contracts. Those need a new potential or
+an explicit comparison. Arbitrary dirty inputs cannot simply be deleted from
+the endpoint sum.
+
+## 3. A matched-translation theorem
+
+Let Gamma_in and Gamma_out be two endpoint charts in the same ambient
+Clifford space. Put K=Gamma_out Gamma_in^-1 and let
+
+    delta(K)=rank(M(K)-I).
+
+This full symplectic defect is conjugacy invariant. It is only an upper
+bound on phi, not a native tariff: B is a submatrix of M-I, giving
+phi(M)<=delta(M). For arbitrary fixed Clifford L and R, set
+
+    Gamma_in'=L Gamma_in R,
+    Gamma_out'=L Gamma_out R.
+
+Then Gamma_out' (Gamma_in')^-1=L K L^-1, and consequently
+
+    |phi(Gamma_out')-phi(Gamma_in')| <= phi(L K L^-1)
+                                               <= delta(K).       (1)
+
+Equation (1) permits arbitrary ambient symplectic L/R, spectator coordinates,
+inverse orientation, phases, and nonorthogonal binary address descriptions.
+It is a theorem about matched endpoints. It does not assert that every such
+translation is legal at every internal addition, has the same native cost,
+or supplies the prescribed global endpoint. Different translations at the
+two ends introduce another operator and are outside (1) until that operator
+is included and paid.
+
+If only one side is translated, the sharper phi(K) or phi(Gamma_in^-1
+Gamma_out) triangle can suffice. The conjugacy-invariant defect is useful
+when both sides vary. No claim is made that a large defect lower-bounds the
+cost of a translation; the inequality is in the opposite direction.
+
+## 4. Intrinsic P104 endpoints and sharpness
+
+The source native proof and compiler specify, for each odd physical root
+label q in F2^9,
+
+    C_q=((1+i)I+(1-i)X_q)/2,
+    F_9=product_j C_ej,
+    M(C_q)=[I q q^t; 0 I],       M(F_9)=[I I;0 I].
+
+The scalar sign of C_q or its inverse does not affect its unsigned
+symplectic action. Since q^t q=1, I+q q^t has kernel exactly span(q) and rank
+8. The 280 intrinsic chart endpoints are therefore:
+
+| Roles | Entrance | Exit | Relative symplectic defect |
+|---|---|---|---:|
+| 104 X | C_q | F_9 | 8 |
+| 104 Y | I | F_9 C_q^-1 | 8 |
+| 72 dirty helpers | I | F_9 | 9 |
+
+Each relative matrix has just the displayed upper-right block, so its defect
+is exactly 8, 8 or 9. Applying (1) separately to every legally retained role
+bounds total signed progress by
+
+    104*8+104*8+72*9=2312.
+
+The original intrinsic endpoints attain equality, proving sharpness for this
+endpoint specification. Inverse-oriented calls have the same absolute cap;
+this is not a proof that the required opposite-oriented physical caller
+exists. Phase/global-sign equality remains a separate implementation
+obligation; unsigned rank checks do not certify it.
+
+For a nondegenerate active space E with column basis V and nonsingular Gram
+G=V^t V, the orthogonal projector is P_E=V G^-1 V^t. It is symmetric,
+idempotent, and has rank dim(E). The quadratic Fourier phase defining C_E
+has polarization 2 x^t P_E y, hence M(C_E)=[I P_E;0 I]. This includes
+alternating nondegenerate spaces, without requiring an orthonormal basis.
+For nested nondegenerate spaces, projector subtraction has rank equal to
+the dimension difference. Tensor-axis embedding into a larger ambient space
+and mixed inverse signs do not alter these defect ranks. The independent
+control uses the alternating plane with basis (011,110) in F2^3, whose
+projector has rank two.
+
+**Changed endpoints can reopen the gate.** If an actual new word leaves every
+Y in F_9 instead of F_9 C_q^-1, its Y cap is nine. With the same X and helper
+endpoints, (1) gives the sharp new total 2416. The old 2312 must not be reused.
+This is an endpoint-dependent corollary, not proof of the concurrent changed-
+consumer construction, its paid mass, source-consuming logical return, or its
+global moment. The complete bill must be recomputed from that actual word.
+
+## 5. Why the canonical chart adapters matter
+
+The native compiler uses a particular `canonical_frame` Gamma_W. It is a
+monomial conjugate of a coordinate C tensor and is not always the intrinsic
+C_W. The published program has **208 rank-zero intrinsic endpoint adapters**:
+104 source-entry adapters and 104 target-exit adapters. They add no positive
+child to the original 2368 invoice. They are distinct from the **208 rank-one
+side-correction transports** used by the later corrected profile.
+
+A fresh canonical-frame census gives:
+
+- 102 X endpoint cocycles: phi=8, full symplectic defect=10;
+- the other two X endpoint cocycles: phi=8, defect=8;
+- all 104 Y endpoint cocycles: phi=8, defect=8;
+- all 72 helper endpoint cocycles: phi=9, defect=9.
+
+Thus applying an intrinsic defect-eight assertion to bare canonical X charts
+would be incorrect. Here is a concrete control at actual port X0, whose
+physical normal is q=386. Let G=Gamma_span(q), F=F_9 and let H be a Hadamard
+symplectic swap on zero-based coordinate 1. A Gaussian-dyadic realization is
+S C S on that bit, up to its explicitly retained global phase; use its actual
+inverse where required. The source-bound matrices give
+
+    phi(F G^-1)=8,  delta(F G^-1)=10,
+    phi(H F G^-1 H^-1)=9.
+
+Choose L=H and R=G^-1 H^-1. Then the matched bare canonical entrance L G R
+is identity, while the exit L F R has phi=9. The claimed per-source cap eight
+would fail for these bare endpoints. This is a legal Clifford endpoint
+example, not a lower-cost P104 supplier or a counterexample to (1).
+
+The missing intrinsic source adapter is A=G C_q^-1. Before translation
+phi(A)=0, but after this translation phi(H A H^-1)=2. Including it restores
+the intended intrinsic cocycle, for which
+
+    phi(H F C_q^-1 H^-1)=8.
+
+One cannot translate the canonical chart and continue donating its old
+rank-zero adapter classification. Every converted wrapper must be evaluated
+in its actual position. Conversely, this does not say every translation makes
+adapters positive: ordinary monomial translations preserve phi.
+
+As an explicit whole-word control, apply the same L=H and R=G^-1 H^-1 to all
+280 canonical charts. Identical-chart additions remain identical-chart
+additions. Reprice each of the 1410 literal transports by its actual
+conjugated B block. The resulting canonical chart progress is 2066 and the
+transport rank mass is 2248, leaving excess182. The original pair was
+2312 and2368, excess56. This particular reduction in paid mass therefore
+worsens the progress deficit. Intrinsic adapters and the side correction
+still require separate repricing; no claim is made that2248 is a complete
+corrected-call bill. The general proof, not this single census, supports the
+intrinsic matched-translation cap.
+
+## 6. Decision and cost consequence
+
+NARROW translation-only attempts to raise progress while keeping the same
+intrinsic endpoint contract. In the role-local common-chart grammar, the
+progress cap used by PR3954/3962 is now justified for the matched translations
+above. Their cost rejection still additionally assumes the corrected mass,
+actual global initial credits/owner endpoint, native rank grammar and volume
+normalization. Arbitrary translations can change the paid profile, so this
+study does not promote those conditional screens to a universal P104 barrier.
+
+Continue a changed Y endpoint, coupled operator-valued encoding, or genuinely
+new word only with its actual endpoint-derived potential and full invoice.
+A feasible first test is the concurrent all-F9 target proposal: verify its
+whole logical return and paid events, then use2416 only if its intrinsic
+endpoint truly matches the displayed changed contract. No selected frontier,
+shared controls or frozen predecessor evidence is changed here.

research/p104-translated-progress/README.md

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+# P104 translated progress: symplectic width and an adapter-sensitive boundary
+
+**Result:** matched Clifford translations cannot raise the persistent progress
+of the unchanged **intrinsic** P104 endpoints above 2312. The proof uses the
+actual native width `phi(M)=rank_F2(B)` of a Clifford symplectic matrix and a
+conjugacy-invariant upper bound from `rank(M-I)`. It applies to role-local
+charts and common-chart scalar additions, with the same translation at both
+ends. It does not assume that translations retain the paid native profile.
+
+The source convention matters. At actual X0, translating the **bare canonical**
+endpoint changes its native width from 8 to 9. Its previously rank-zero source
+adapter then costs width 2. Including the intrinsic adapter restores width 8.
+This is a concrete missing-hypothesis control, not a supplier improvement.
+The 208 rank-zero intrinsic adapters are distinct from the later 208 rank-one
+side-correction transports.
+
+The theorem is reusable after a real endpoint change. If every Y ends at F9
+instead of F9*Cq^-1, the corresponding sharp progress cap becomes **2416**.
+The old 2312 must not be applied to that changed contract. Any such new word
+still needs its actual logical return, complete bill and global caller proof.
+
+[PROOF.md](PROOF.md) contains the general matrix proof, nonorthogonal and
+alternating residual treatment, exact source binding, counterexample and
+scope limits. [RESULT.json](RESULT.json) contains the finite receipt. The
+source-bound whole canonical Hadamard control has native mass 2248 and
+progress 2066, worsening their difference from 56 to 182; intrinsic adapters
+and side correction remain separately chargeable. No complete changed-call
+invoice or new kappa is claimed. The selected frontier is unchanged.
+
+## Reproduction
+
+Python 3.12, standard library only, from the repository root:
+
+```sh
+python3 -B research/p104-translated-progress/verify.py
+python3 -O -B research/p104-translated-progress/independent.py
+python3 -O -B -m unittest discover -s research/p104-translated-progress -p test_verify.py -v
+python3 -B scripts/research_package.py --package research/p104-translated-progress
+```
+
+The primary checker authenticates nine canonical inputs (including the retained compiler notice and license), reconstructs all
+14400 root pairings and the 104 physical ports, binds all 1410 original
+transport widths, checks all 280 canonical endpoint cocycles, and recomputes
+the translated control. It imports the pinned native compiler only to
+instantiate its actual canonical-frame convention; it does not import the
+channel generator or treat that compiler as an independent checker.
+
+The separate checker imports no producer or primary verifier. Dense binary
+arithmetic checks all 518400 products of Sp(4,2), 2160 matched translations,
+and the actual 18-by-18 counterexample/adapter matrices. Exact Gaussian
+rational operators independently recover six Cq/inverse Pauli actions.
+Six unittest methods include seven actual optimized-CLI receipt/source faults
+(each preceded by a passing private-fixture baseline) and mathematical controls for unmatched endpoints, even labels, changed
+outputs and non-invariant native width. Typed receipts and duplicate JSON keys
+fail closed. All commands support a read-only, Git-free source export; test
+mutations affect only disposable private copies.
+
+These checks support the scoped proof. Unsigned symplectic ranks do not
+certify phases, the complete physical channel, signed-floor/precision or
+all-size admission. The old native phase checks retain their own source scope.
+
+## Sources, credit and overlap
+
+Baseline: canonical main `20ecf2891d80b5acf18d980b2164fb44ea7faa90`, priority
+revision 6. [SOURCES.json](SOURCES.json) contains exact base/blob/SHA256 pins;
+all replay inputs are available in canonical history. Pending PRs motivate
+the question but are not imported as hidden Git dependencies.
+
+- Rohan Arun / [PR1972](https://github.com/CrocSwap/integer-mult-bounds/pull/1972)
+  and Adam Gall / PR1940 supply the scalar/frame word and schedule.
+- [PR1991](https://github.com/CrocSwap/integer-mult-bounds/pull/1991)
+  supplies the native realization and intrinsic adapters, prepared for Chafik Boukhalfa. Its compiler credit
+  remains with icekylinx's frame-sa504 work; physical-label and independent
+  interpreter credit remains with the predecessor chain recorded there.
+- Douglas Colkitt and the credited authors of `notes/endpoint-gauge-complex.tex`
+  and `notes/copied-centers-complex.tex` supply the residual normal form and
+  actual translated/exterior caller semantics.
+- [PR3954](https://github.com/CrocSwap/integer-mult-bounds/pull/3954), head
+  `f1e2839819959cd0534d008be91ed5b79ec55a6a`, and
+  [PR3962](https://github.com/CrocSwap/integer-mult-bounds/pull/3962), head
+  `f7572974ac7fbcd21fd71c73b64bca0c8487aed6`, leave the conditional progress
+  implication audited here. Their incidence and full-gate results are not
+  claimed anew.
+- [PR3882](https://github.com/CrocSwap/integer-mult-bounds/pull/3882),
+  [PR3894](https://github.com/CrocSwap/integer-mult-bounds/pull/3894),
+  [PR3940](https://github.com/CrocSwap/integer-mult-bounds/pull/3940),
+  [PR3961](https://github.com/CrocSwap/integer-mult-bounds/pull/3961) and
+  [PR3965](https://github.com/CrocSwap/integer-mult-bounds/pull/3965) own the
+  correction, encoded boundary and copy-lifetime contexts. ToikY's
+  [PR3966](https://github.com/CrocSwap/integer-mult-bounds/pull/3966) concerns
+  phase-uniform gauges, a different obligation.
+
+Clear1nsight, with substantial OpenAI Codex multi-agent assistance. Apache-2.0.
+No literature-priority, proof-assistant or independent human-review claim.
+
+## Open next questions
+
+1. **Changed intrinsic target endpoint:** verify the proposed all-F9 target
+   word and price its complete correction/consumer. The first step is its
+   all-role physical boundary; only then use the 2416 cap in a new global bill.
+2. **Coupled operator-valued charts:** determine whether a cross-role encoding
+   admits a useful replacement for this role-local sum. Start with one actual
+   adjacent pair and its complete transition, not a free abstract mixing.
+3. **Translation with changed native profile:** price every conjugated child,
+   intrinsic adapter, exterior and final wrapper before comparing costs.
+   The source-bound X0 control shows why old rank-zero classifications cannot
+   simply be carried through a new Clifford conjugation.
+
+Decision: NARROW matched-translation-only progress searches under unchanged
+intrinsic endpoints; retain changed endpoints and coupled algorithms as
+explicitly different research directions. Structured next steps are in
+[TASK.json](TASK.json).